<?xml version="1.0" encoding="UTF-8"?><rss xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:content="http://purl.org/rss/1.0/modules/content/" xmlns:atom="http://www.w3.org/2005/Atom" version="2.0"><channel><title><![CDATA[Makon AI Blog]]></title><description><![CDATA[Makon AI is your always-on SAT tutor. 5,000+ practice questions, AI-powered study plans, mock tests, and Po — your personal AI tutor — to help you reach 1600.]]></description><link>http://github.com/dylang/node-rss</link><generator>GatsbyJS</generator><lastBuildDate>Mon, 04 May 2026 23:33:59 GMT</lastBuildDate><item><title><![CDATA[Digital SAT Format Explained: Sections, Timing, Modules & Adaptive Scoring (2026 Guide)]]></title><description><![CDATA[Everything on the Digital SAT — the two sections, four adaptive modules, exact timing per part, question counts, the Bluebook app, built-in tools, and how the scoring really works.]]></description><link>https://www.makon.ai/blog/digital-sat-format</link><guid isPermaLink="false">https://www.makon.ai/blog/digital-sat-format</guid><pubDate>Mon, 04 May 2026 00:00:00 GMT</pubDate><content:encoded>&lt;p&gt;In March 2024 the SAT went fully digital, ending a 100-year run of paper-and-pencil exams. The redesigned test isn’t just the same exam on a screen — it’s shorter, adaptive, and uses the College Board’s &lt;strong&gt;Bluebook&lt;/strong&gt; app as the testing platform. If you’re prepping for the SAT in 2026 or 2027, the &lt;em&gt;only&lt;/em&gt; version you’ll take is the Digital SAT.&lt;/p&gt;
&lt;p&gt;This guide covers everything about the Digital SAT format: how it’s structured, how the adaptive modules work, exactly how long each section takes, what tools you have on screen, and how the scoring actually shakes out. By the end, you’ll know the test inside and out — which is the foundation for prepping efficiently.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Quick references:&lt;/strong&gt; check our &lt;a href=&quot;/sat-score-calculator&quot;&gt;free Digital SAT score calculator&lt;/a&gt;, the &lt;a href=&quot;/blog/sat-math-formulas&quot;&gt;SAT math formula sheet&lt;/a&gt;, and the &lt;a href=&quot;/blog/sat-test-dates-2026&quot;&gt;2026 test date calendar&lt;/a&gt;.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;the-digital-sat-at-a-glance&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-digital-sat-at-a-glance&quot; aria-label=&quot;the digital sat at a glance permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The Digital SAT at a glance&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Spec&lt;/th&gt;
&lt;th&gt;Detail&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Total testing time&lt;/td&gt;
&lt;td&gt;2 hours 14 minutes&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Break&lt;/td&gt;
&lt;td&gt;10 minutes (between Reading &amp;#x26; Writing and Math)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Total questions&lt;/td&gt;
&lt;td&gt;98&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Sections&lt;/td&gt;
&lt;td&gt;2 (Reading &amp;#x26; Writing, Math)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Modules per section&lt;/td&gt;
&lt;td&gt;2 (adaptive)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Score range&lt;/td&gt;
&lt;td&gt;400 – 1600 (200 – 800 per section)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Calculator&lt;/td&gt;
&lt;td&gt;Allowed on every Math question&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Platform&lt;/td&gt;
&lt;td&gt;Bluebook (College Board’s testing app)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;That’s shorter than the legacy SAT (3 hours of testing, 154 questions) and roughly the same length as a feature film.&lt;/p&gt;
&lt;h2 id=&quot;the-two-sections-of-the-digital-sat&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-two-sections-of-the-digital-sat&quot; aria-label=&quot;the two sections of the digital sat permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The two sections of the Digital SAT&lt;/h2&gt;
&lt;p&gt;Unlike the old SAT — which had Reading, Writing, and Math as separate sections — the Digital SAT bundles reading and writing into a single section.&lt;/p&gt;
&lt;h3 id=&quot;section-1-reading--writing-rw&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#section-1-reading--writing-rw&quot; aria-label=&quot;section 1 reading  writing rw permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Section 1: Reading &amp;#x26; Writing (R&amp;#x26;W)&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;54 questions&lt;/strong&gt; total (across two modules)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;64 minutes&lt;/strong&gt; total (32 minutes per module)&lt;/li&gt;
&lt;li&gt;Each question is tied to a &lt;strong&gt;short passage&lt;/strong&gt; (25 – 150 words) — much shorter than the old SAT’s long passages&lt;/li&gt;
&lt;li&gt;One question per passage (no more 10-question chunks on a single text)&lt;/li&gt;
&lt;li&gt;Mix of literature, history, social studies, science, and the humanities&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;The four broad question categories on Reading &amp;#x26; Writing:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Craft and Structure&lt;/strong&gt; (~28%): vocabulary in context, text purpose, cross-text connections&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Information and Ideas&lt;/strong&gt; (~26%): central ideas, details, evidence, inferences&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Standard English Conventions&lt;/strong&gt; (~26%): grammar, punctuation, sentence structure&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Expression of Ideas&lt;/strong&gt; (~20%): rhetorical synthesis, transitions&lt;/li&gt;
&lt;/ol&gt;
&lt;h3 id=&quot;section-2-math&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#section-2-math&quot; aria-label=&quot;section 2 math permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Section 2: Math&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;44 questions&lt;/strong&gt; total (across two modules)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;70 minutes&lt;/strong&gt; total (35 minutes per module)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Calculator allowed on every question&lt;/strong&gt; (no more no-calc section)&lt;/li&gt;
&lt;li&gt;Built-in graphing calculator (Desmos) is on screen — you can also bring your own approved one&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;The four math content areas:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Algebra&lt;/strong&gt; (~35%): linear equations, systems, inequalities&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Advanced Math&lt;/strong&gt; (~35%): quadratics, polynomials, exponents, function notation&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Problem-Solving and Data Analysis&lt;/strong&gt; (~15%): ratios, percentages, statistics, probability&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Geometry and Trigonometry&lt;/strong&gt; (~15%): area, volume, circles, right triangles, basic trig&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;For the formulas you’re given on test day plus the ones you have to memorize, see our &lt;a href=&quot;/blog/sat-math-formulas&quot;&gt;SAT math formula sheet&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id=&quot;the-four-modules-how-time-is-broken-down&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-four-modules-how-time-is-broken-down&quot; aria-label=&quot;the four modules how time is broken down permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The four modules: how time is broken down&lt;/h2&gt;
&lt;p&gt;Each section is split into two modules of equal length. Here’s the exact time breakdown:&lt;/p&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Module&lt;/th&gt;
&lt;th&gt;Questions&lt;/th&gt;
&lt;th&gt;Time&lt;/th&gt;
&lt;th&gt;Type&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Reading &amp;#x26; Writing — Module 1&lt;/td&gt;
&lt;td&gt;27&lt;/td&gt;
&lt;td&gt;32 minutes&lt;/td&gt;
&lt;td&gt;Static (same for everyone)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Reading &amp;#x26; Writing — Module 2&lt;/td&gt;
&lt;td&gt;27&lt;/td&gt;
&lt;td&gt;32 minutes&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;Adaptive&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;10-minute break&lt;/td&gt;
&lt;td&gt;—&lt;/td&gt;
&lt;td&gt;10 minutes&lt;/td&gt;
&lt;td&gt;—&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Math — Module 1&lt;/td&gt;
&lt;td&gt;22&lt;/td&gt;
&lt;td&gt;35 minutes&lt;/td&gt;
&lt;td&gt;Static (same for everyone)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Math — Module 2&lt;/td&gt;
&lt;td&gt;22&lt;/td&gt;
&lt;td&gt;35 minutes&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;Adaptive&lt;/strong&gt;&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;That’s &lt;strong&gt;2 hours 14 minutes&lt;/strong&gt; of pure testing time. With the 10-minute break it’s about 2:24. Plus check-in (~30 minutes) at the test center, plan for ~3 hours from arrival to leaving.&lt;/p&gt;
&lt;h2 id=&quot;how-adaptive-routing-actually-works&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-adaptive-routing-actually-works&quot; aria-label=&quot;how adaptive routing actually works permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How adaptive routing actually works&lt;/h2&gt;
&lt;p&gt;This is the single most-misunderstood part of the Digital SAT. The test is &lt;strong&gt;section-adaptive&lt;/strong&gt;, not question-adaptive — your performance on Module 1 determines which Module 2 you get, and the difficulty of those questions then affects your scoring ceiling.&lt;/p&gt;
&lt;h3 id=&quot;the-mechanics&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-mechanics&quot; aria-label=&quot;the mechanics permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The mechanics&lt;/h3&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Module 1 of each section is the same for everyone.&lt;/strong&gt; Mixed difficulty: a few easy questions, mostly medium, a few hard. This module establishes your skill baseline.&lt;/li&gt;
&lt;li&gt;After you finish Module 1, the test scores it instantly.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Module 2 routes you&lt;/strong&gt; to one of two paths:
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Higher path:&lt;/strong&gt; harder questions, with a higher score ceiling.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Lower path:&lt;/strong&gt; easier questions, with a lower score ceiling.&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;The threshold for the higher path isn’t public, but in practice you usually need to get &lt;strong&gt;about 70% or more&lt;/strong&gt; of Module 1 right to route up.&lt;/li&gt;
&lt;/ol&gt;
&lt;h3 id=&quot;what-this-means-for-your-score&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#what-this-means-for-your-score&quot; aria-label=&quot;what this means for your score permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What this means for your score&lt;/h3&gt;
&lt;p&gt;If you ace the higher Module 2, you can score the full 800. If you’re routed to the lower Module 2 — even with a perfect score on it — your &lt;strong&gt;maximum scaled score for that section is roughly 600&lt;/strong&gt;.&lt;/p&gt;
&lt;p&gt;That’s why two students can both miss 5 questions but earn very different scores: one missed mostly Module 1 questions and got routed down, the other missed Module 2 questions on the higher path.&lt;/p&gt;
&lt;h3 id=&quot;practical-consequence&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#practical-consequence&quot; aria-label=&quot;practical consequence permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Practical consequence&lt;/h3&gt;
&lt;p&gt;Spend serious time on Module 1. Every question there matters more than a question of the same apparent difficulty in Module 2 — because it changes your ceiling. Our &lt;a href=&quot;/sat-score-calculator&quot;&gt;Digital SAT score calculator&lt;/a&gt; factors this routing into its estimate.&lt;/p&gt;
&lt;h2 id=&quot;bluebook-the-testing-app&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#bluebook-the-testing-app&quot; aria-label=&quot;bluebook the testing app permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Bluebook: the testing app&lt;/h2&gt;
&lt;p&gt;The Digital SAT runs in &lt;strong&gt;Bluebook&lt;/strong&gt;, College Board’s free testing app. You install it on a personal device (laptop or tablet) and log in with your College Board account on test day.&lt;/p&gt;
&lt;p&gt;Compatible devices:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Mac, Windows, ChromeBook&lt;/li&gt;
&lt;li&gt;iPad (9.7-inch or larger), Windows tablet&lt;/li&gt;
&lt;li&gt;School-managed devices (district provides them)&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;You can practice on Bluebook &lt;em&gt;before&lt;/em&gt; test day — College Board provides free official practice tests inside the app, and they’re scored the same way as the real test. &lt;strong&gt;Do this&lt;/strong&gt;. Test day is not the time to fight the interface.&lt;/p&gt;
&lt;h2 id=&quot;built-in-tools-you-have-on-screen&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#built-in-tools-you-have-on-screen&quot; aria-label=&quot;built in tools you have on screen permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Built-in tools you have on screen&lt;/h2&gt;
&lt;p&gt;The digital format comes with several utilities baked into Bluebook:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Desmos graphing calculator&lt;/strong&gt; — embedded, available on every Math question. You can also bring your own approved calculator if you prefer (TI-84, Casio fx-9750GIII, etc.).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Reference sheet&lt;/strong&gt; — the 13 official math formulas (areas, volumes, special right triangles) are one click away on every Math question. See our &lt;a href=&quot;/blog/sat-math-formulas&quot;&gt;math formula sheet guide&lt;/a&gt; for the formulas the SAT does &lt;em&gt;not&lt;/em&gt; give you.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Mark for review (flag)&lt;/strong&gt; — flag any question to revisit later within the same module. The flag persists until you submit that module.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Annotation tool&lt;/strong&gt; — highlight passage text in Reading &amp;#x26; Writing for reference.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Question navigator&lt;/strong&gt; — see all 27 (R&amp;#x26;W) or 22 (Math) questions at once, jump around within a module.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Cross-out tool&lt;/strong&gt; — eliminate wrong multiple-choice options directly on screen.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Built-in timer&lt;/strong&gt; — counts down for each module. You can hide it if it’s stressful.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Important: you &lt;strong&gt;cannot move between modules&lt;/strong&gt;. Once you submit Module 1, Module 2 starts and there’s no going back.&lt;/p&gt;
&lt;h2 id=&quot;question-types-youll-see&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#question-types-youll-see&quot; aria-label=&quot;question types youll see permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Question types you’ll see&lt;/h2&gt;
&lt;h3 id=&quot;reading--writing&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#reading--writing&quot; aria-label=&quot;reading  writing permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Reading &amp;#x26; Writing&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Multiple choice (4 options)&lt;/strong&gt; — every R&amp;#x26;W question is multiple choice. There’s no essay on the Digital SAT.&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&quot;math&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#math&quot; aria-label=&quot;math permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Math&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Multiple choice (4 options)&lt;/strong&gt; — most questions&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Student-produced response (grid-in)&lt;/strong&gt; — about 25% of Math questions ask you to type a numeric answer. Decimals, fractions, and negatives are allowed; you don’t have to bubble in.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;There are no “select all that apply” questions, no drag-and-drop, no audio. Every question is one answer, scored as right or wrong (no partial credit, no penalty for wrong answers — guess on everything).&lt;/p&gt;
&lt;h2 id=&quot;how-the-digital-sat-is-scored&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-the-digital-sat-is-scored&quot; aria-label=&quot;how the digital sat is scored permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How the Digital SAT is scored&lt;/h2&gt;
&lt;h3 id=&quot;the-200--800-section-scale&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-200--800-section-scale&quot; aria-label=&quot;the 200  800 section scale permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The 200 – 800 section scale&lt;/h3&gt;
&lt;p&gt;Each section (Reading &amp;#x26; Writing and Math) is scored on a 200 – 800 scale. They sum to your &lt;strong&gt;total score, 400 – 1600&lt;/strong&gt; — the same range as the legacy paper SAT, so an 1100 means the same thing on either format.&lt;/p&gt;
&lt;h3 id=&quot;from-raw-to-scaled&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#from-raw-to-scaled&quot; aria-label=&quot;from raw to scaled permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;From raw to scaled&lt;/h3&gt;
&lt;p&gt;Your raw score (number of questions you got right) is converted to a scaled score using:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;The Module 2 path you were on&lt;/strong&gt; (higher or lower)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Statistical equating&lt;/strong&gt; — the College Board adjusts for the specific difficulty of the test form you took, so two students with the same raw correct count on different test dates get equivalent scaled scores.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;The exact conversion table isn’t publicly published, but the rough shape:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Higher Module 2 path, 30 / 27 raw correct in R&amp;#x26;W → ~700 R&amp;#x26;W&lt;/li&gt;
&lt;li&gt;Higher path, 35 / 44 raw correct in Math → ~700 Math&lt;/li&gt;
&lt;li&gt;Lower path, even 27/27 + 27/27 perfect raw → R&amp;#x26;W cap ~600&lt;/li&gt;
&lt;li&gt;Lower path, 22/22 perfect raw → Math cap ~600&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&quot;wrong-answers-dont-cost-you&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#wrong-answers-dont-cost-you&quot; aria-label=&quot;wrong answers dont cost you permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Wrong answers don’t cost you&lt;/h3&gt;
&lt;p&gt;There’s &lt;strong&gt;no penalty for guessing&lt;/strong&gt;. Always answer every question, even if you’re completely guessing. A 25% chance at a multiple-choice question is infinitely better than a guaranteed zero.&lt;/p&gt;
&lt;h3 id=&quot;score-release&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#score-release&quot; aria-label=&quot;score release permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Score release&lt;/h3&gt;
&lt;p&gt;Scores come out about &lt;strong&gt;two weeks after test day&lt;/strong&gt; — much faster than the old paper SAT. You’ll get an email when they post.&lt;/p&gt;
&lt;p&gt;For score interpretation, college tier mapping, and what counts as a “good” score, see &lt;a href=&quot;/blog/good-sat-score&quot;&gt;What’s a Good SAT Score in 2026?&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id=&quot;digital-sat-vs-legacy-paper-sat&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#digital-sat-vs-legacy-paper-sat&quot; aria-label=&quot;digital sat vs legacy paper sat permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Digital SAT vs. legacy paper SAT&lt;/h2&gt;
&lt;p&gt;For context, here’s what changed when the test went digital:&lt;/p&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Feature&lt;/th&gt;
&lt;th&gt;Legacy paper SAT&lt;/th&gt;
&lt;th&gt;Digital SAT&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Total testing time&lt;/td&gt;
&lt;td&gt;3:00&lt;/td&gt;
&lt;td&gt;2:14&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Total questions&lt;/td&gt;
&lt;td&gt;154&lt;/td&gt;
&lt;td&gt;98&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Sections&lt;/td&gt;
&lt;td&gt;4 (Reading, Writing, Math No-Calc, Math Calc)&lt;/td&gt;
&lt;td&gt;2 (R&amp;#x26;W, Math)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Adaptive&lt;/td&gt;
&lt;td&gt;No&lt;/td&gt;
&lt;td&gt;Yes (section-adaptive)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Calculator on math&lt;/td&gt;
&lt;td&gt;Half the time&lt;/td&gt;
&lt;td&gt;Every question&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Question format&lt;/td&gt;
&lt;td&gt;Bubble sheet&lt;/td&gt;
&lt;td&gt;Bluebook app&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Reading passages&lt;/td&gt;
&lt;td&gt;Long (500–700 words)&lt;/td&gt;
&lt;td&gt;Short (25–150 words)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Score release&lt;/td&gt;
&lt;td&gt;2–4 weeks&lt;/td&gt;
&lt;td&gt;~2 weeks&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Score range&lt;/td&gt;
&lt;td&gt;400 – 1600&lt;/td&gt;
&lt;td&gt;400 – 1600&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;Same scale, different test. Colleges treat scores from both as equivalent for admissions.&lt;/p&gt;
&lt;h2 id=&quot;how-to-prepare-for-the-format&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-to-prepare-for-the-format&quot; aria-label=&quot;how to prepare for the format permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How to prepare for the format&lt;/h2&gt;
&lt;p&gt;A few format-specific moves that pay off:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Take the official practice tests inside Bluebook&lt;/strong&gt;, not on paper. The interface, pacing, and tools are different — practicing on paper teaches the wrong muscle memory.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Get good at the Desmos calculator.&lt;/strong&gt; Most students under-use it. Knowing how to graph an equation, find intersections, and solve systems graphically can save you 30+ seconds per question.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Lean into Module 1.&lt;/strong&gt; As covered above, Module 1 questions disproportionately affect your score because they determine your routing path. Don’t treat it like a warm-up.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Practice timing inside the modules&lt;/strong&gt;, not across the whole section. You can’t move time from Module 1 to Module 2 — they’re separate clocks.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Use the flag liberally.&lt;/strong&gt; If a question is taking more than 90 seconds, flag it and come back. Module 2 in R&amp;#x26;W gives you 32 minutes for 27 questions — that’s 71 seconds per question on average.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;For practice questions that match the Digital SAT’s difficulty distribution and adaptive routing, our AI tutor on &lt;a href=&quot;https://app.makon.ai&quot;&gt;Makon AI&lt;/a&gt; walks you through every wrong answer and adapts to the topics you keep missing.&lt;/p&gt;
&lt;h2 id=&quot;faqs&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#faqs&quot; aria-label=&quot;faqs permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;FAQs&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;How long is the Digital SAT?&lt;/strong&gt;
2 hours and 14 minutes of testing time, plus a 10-minute break. Total time at the test center, including check-in, is about 3 hours.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;How long is each section of the SAT?&lt;/strong&gt;
Reading &amp;#x26; Writing: 64 minutes (two 32-minute modules). Math: 70 minutes (two 35-minute modules). 10-minute break in between.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;How many questions are on the Digital SAT?&lt;/strong&gt;
98 total — 54 in Reading &amp;#x26; Writing (27 + 27 across two modules) and 44 in Math (22 + 22). A handful of those are unscored pretest questions used to calibrate future exams.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is the Digital SAT adaptive?&lt;/strong&gt;
Yes — it’s section-adaptive. Module 1 of each section is the same for everyone, and your performance there determines whether Module 2 routes you to the harder or easier path.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is the Digital SAT all multiple choice?&lt;/strong&gt;
Reading &amp;#x26; Writing is 100% multiple choice. Math is mostly multiple choice with about 11 of the 44 questions being student-produced response (you type a numeric answer).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is there a calculator on the Digital SAT?&lt;/strong&gt;
Yes — calculator is allowed on every Math question. Bluebook includes a built-in Desmos graphing calculator, and you can also bring your own approved calculator.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is there an essay on the Digital SAT?&lt;/strong&gt;
No. The optional essay was eliminated. There’s no writing-from-scratch requirement.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Can I move between modules?&lt;/strong&gt;
No. Once you submit Module 1, you can’t go back. Within a module, though, you can navigate freely between questions and use the “mark for review” flag.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;What math is on the Digital SAT?&lt;/strong&gt;
Algebra (~35%), Advanced Math / quadratics &amp;#x26; functions (~35%), Problem-Solving and Data Analysis (~15%), Geometry and Trigonometry (~15%). Full breakdown in &lt;a href=&quot;/blog/sat-math-formulas&quot;&gt;our math formula guide&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Do colleges treat the Digital SAT the same as the old SAT?&lt;/strong&gt;
Yes. The score scale is identical (400 – 1600) and College Board has confirmed all admissions offices treat the scores as equivalent.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;When can I take the Digital SAT?&lt;/strong&gt;
Multiple dates per year. See our full &lt;a href=&quot;/blog/sat-test-dates-2026&quot;&gt;2026 SAT test dates calendar&lt;/a&gt;.&lt;/p&gt;</content:encoded></item><item><title><![CDATA[How Long Is the SAT? (Digital SAT 2026 Timing Guide)]]></title><description><![CDATA[The Digital SAT runs 2 hours 14 minutes of testing time, plus a 10-minute break — but plan for ~3 hours total at the test center. Here's the full timing breakdown by section, module, and check-in.]]></description><link>https://www.makon.ai/blog/how-long-is-the-sat</link><guid isPermaLink="false">https://www.makon.ai/blog/how-long-is-the-sat</guid><pubDate>Mon, 04 May 2026 00:00:00 GMT</pubDate><content:encoded>&lt;p&gt;The short answer: &lt;strong&gt;the Digital SAT takes 2 hours and 14 minutes of testing time&lt;/strong&gt;, plus a 10-minute break. Add check-in and you should plan for &lt;strong&gt;about 3 hours total&lt;/strong&gt; at the test center.&lt;/p&gt;
&lt;p&gt;That&apos;s 46 minutes shorter than the legacy paper SAT (which was 3 hours straight, no calculator on half of it). The digital version is genuinely faster — fewer questions, no separate no-calculator math section, and the adaptive scoring means College Board can collect the same precision in less time.&lt;/p&gt;
&lt;p&gt;But &quot;2:14 of testing&quot; is the headline number, not the practical one. To plan your test day correctly, you need the per-section, per-module breakdown — plus all the time around the test you can&apos;t skip. Here&apos;s the complete timing.&lt;/p&gt;
&lt;h2 id=&quot;the-digital-sat-timing-at-a-glance&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-digital-sat-timing-at-a-glance&quot; aria-label=&quot;the digital sat timing at a glance permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The Digital SAT timing at a glance&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Phase&lt;/th&gt;
&lt;th&gt;Duration&lt;/th&gt;
&lt;th&gt;Notes&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Check-in&lt;/td&gt;
&lt;td&gt;~30 min&lt;/td&gt;
&lt;td&gt;Plan to arrive at the time on your ticket, not 5 minutes before&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Reading &amp;#x26; Writing Module 1&lt;/td&gt;
&lt;td&gt;32 min&lt;/td&gt;
&lt;td&gt;27 questions, mixed difficulty&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Reading &amp;#x26; Writing Module 2&lt;/td&gt;
&lt;td&gt;32 min&lt;/td&gt;
&lt;td&gt;27 questions, &lt;strong&gt;adaptive&lt;/strong&gt; difficulty&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Break&lt;/td&gt;
&lt;td&gt;10 min&lt;/td&gt;
&lt;td&gt;Mandatory; can&apos;t skip to finish faster&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Math Module 1&lt;/td&gt;
&lt;td&gt;35 min&lt;/td&gt;
&lt;td&gt;22 questions, mixed difficulty&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Math Module 2&lt;/td&gt;
&lt;td&gt;35 min&lt;/td&gt;
&lt;td&gt;22 questions, &lt;strong&gt;adaptive&lt;/strong&gt; difficulty&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Wrap-up + dismissal&lt;/td&gt;
&lt;td&gt;~10 min&lt;/td&gt;
&lt;td&gt;Submission, device check, dismissal&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Total in-test&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;2:14&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Just the four modules&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Total with break&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;2:24&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Modules + the one break&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Total at test center&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;~3:00&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;Add check-in + dismissal&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Timing note:&lt;/strong&gt; the timer inside Bluebook (the testing app) only counts down during the modules themselves. Time spent reading instructions, transitioning between modules, or on break does not eat into your testing time.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;per-section-breakdown&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#per-section-breakdown&quot; aria-label=&quot;per section breakdown permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Per-section breakdown&lt;/h2&gt;
&lt;h3 id=&quot;reading--writing--64-minutes-total&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#reading--writing--64-minutes-total&quot; aria-label=&quot;reading  writing  64 minutes total permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Reading &amp;#x26; Writing — 64 minutes total&lt;/h3&gt;
&lt;p&gt;The R&amp;#x26;W section is &lt;strong&gt;two modules of 32 minutes each&lt;/strong&gt;, back-to-back, with no break between them. That&apos;s 54 questions over 1 hour 4 minutes.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Module 1&lt;/strong&gt; — 27 questions in 32 minutes (~71 sec/question). Mixed difficulty, identical for everyone.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Module 2&lt;/strong&gt; — 27 questions in 32 minutes. Difficulty depends on how you did in Module 1: harder questions if you scored well, easier ones if you didn&apos;t.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Each R&amp;#x26;W question is a short passage (~25–150 words) followed by a single multiple-choice question. You only see one question at a time — no flipping back through a long reading passage to answer 5 sub-questions.&lt;/p&gt;
&lt;h3 id=&quot;math--70-minutes-total&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#math--70-minutes-total&quot; aria-label=&quot;math  70 minutes total permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Math — 70 minutes total&lt;/h3&gt;
&lt;p&gt;After the 10-minute break, you get &lt;strong&gt;two Math modules of 35 minutes each&lt;/strong&gt;, also back-to-back. 44 questions in 1 hour 10 minutes.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Module 1&lt;/strong&gt; — 22 questions in 35 minutes (~95 sec/question). Mixed difficulty.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Module 2&lt;/strong&gt; — 22 questions in 35 minutes. Adaptive routing same as R&amp;#x26;W.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Math has slightly more time per question than R&amp;#x26;W because most problems require setup + computation. The built-in Desmos graphing calculator is available on every question — there&apos;s no longer a &quot;no-calculator&quot; section like the paper SAT had.&lt;/p&gt;
&lt;h2 id=&quot;how-long-is-the-break-on-the-digital-sat&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-long-is-the-break-on-the-digital-sat&quot; aria-label=&quot;how long is the break on the digital sat permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How long is the break on the Digital SAT?&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;10 minutes.&lt;/strong&gt; That&apos;s the only break in the test, between the Reading &amp;#x26; Writing section and the Math section.&lt;/p&gt;
&lt;p&gt;You can:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Use the bathroom&lt;/li&gt;
&lt;li&gt;Eat a snack from your bag&lt;/li&gt;
&lt;li&gt;Drink water&lt;/li&gt;
&lt;li&gt;Stretch&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;You cannot:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Use your phone (it has to stay off and stowed)&lt;/li&gt;
&lt;li&gt;Discuss the test with anyone&lt;/li&gt;
&lt;li&gt;Leave the test center&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;The break starts when the proctor announces it, not when you finish R&amp;#x26;W early. If you finish your R&amp;#x26;W modules in 60 minutes instead of 64, you don&apos;t get a longer break — the test center stays on its proctor-controlled clock.&lt;/p&gt;
&lt;h2 id=&quot;check-in-arrive-30-minutes-early&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#check-in-arrive-30-minutes-early&quot; aria-label=&quot;check in arrive 30 minutes early permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Check-in: arrive 30 minutes early&lt;/h2&gt;
&lt;p&gt;Your admission ticket lists a &quot;report by&quot; time. Hit that time, not the start time. Late arrivals are typically turned away.&lt;/p&gt;
&lt;p&gt;Check-in includes:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;ID + ticket verification&lt;/strong&gt; at the door&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Device check&lt;/strong&gt; — proctor confirms Bluebook is installed, fully updated, and your device is at adequate battery (laptops should have 3+ hours of charge)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Seat assignment&lt;/strong&gt; — you&apos;ll be told where to sit&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Test Preview activation&lt;/strong&gt; — Bluebook unlocks the live test. You&apos;ll need to enter the start code the proctor reads aloud&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Instructions read aloud&lt;/strong&gt; by the proctor (~5 minutes)&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;If 100 students are testing, check-in plus instructions can stretch to 35–40 minutes. Plan for it.&lt;/p&gt;
&lt;h2 id=&quot;how-long-does-it-take-to-get-your-sat-score&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-long-does-it-take-to-get-your-sat-score&quot; aria-label=&quot;how long does it take to get your sat score permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How long does it take to get your SAT score?&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;About two weeks.&lt;/strong&gt; College Board releases Digital SAT scores &lt;strong&gt;roughly 14 days after test day&lt;/strong&gt;. Some test administrations release in 10 days, some take 21 — but two weeks is the planning rule.&lt;/p&gt;
&lt;p&gt;Your scores arrive through three channels:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Your College Board account&lt;/strong&gt; at satsuite.collegeboard.org (the first place they appear)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Email notification&lt;/strong&gt; sent the same day&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Score report sent to colleges&lt;/strong&gt; — about 7–10 days after your account release, depending on whether you selected score recipients&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;If you selected free score recipients during registration, those scores send automatically. Otherwise you can pay to send them later.&lt;/p&gt;
&lt;h2 id=&quot;how-long-is-the-sat-compared-to-the-act-and-psat&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-long-is-the-sat-compared-to-the-act-and-psat&quot; aria-label=&quot;how long is the sat compared to the act and psat permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How long is the SAT compared to the ACT and PSAT?&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Test&lt;/th&gt;
&lt;th&gt;Testing time&lt;/th&gt;
&lt;th&gt;Total at test center&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Digital SAT&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;2:14&lt;/td&gt;
&lt;td&gt;~3:00&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;PSAT/NMSQT&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;2:14&lt;/td&gt;
&lt;td&gt;~3:00&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;ACT&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;2:55 (no Writing) / 3:35 (with Writing)&lt;/td&gt;
&lt;td&gt;~3:30 / ~4:30&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;strong&gt;Legacy paper SAT&lt;/strong&gt; (retired)&lt;/td&gt;
&lt;td&gt;3:00&lt;/td&gt;
&lt;td&gt;~4:00&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;The Digital SAT is now the &lt;strong&gt;shortest&lt;/strong&gt; of the major standardized college admissions tests. The PSAT is identical in length because it&apos;s the same test format, just at lower difficulty.&lt;/p&gt;
&lt;p&gt;The ACT remains noticeably longer because of its Science section (no SAT equivalent) and longer reading passages. If timing/stamina is a major concern for you, the SAT is the easier test to physically endure.&lt;/p&gt;
&lt;h2 id=&quot;what-if-i-run-out-of-time-on-a-module&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#what-if-i-run-out-of-time-on-a-module&quot; aria-label=&quot;what if i run out of time on a module permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What if I run out of time on a module?&lt;/h2&gt;
&lt;p&gt;You can&apos;t pause or extend a module. When the timer hits zero, the module submits whatever you have — answered, flagged, or blank. Strategy:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Never leave a question blank.&lt;/strong&gt; There&apos;s no penalty for guessing on the Digital SAT. If you&apos;re running out of time, fill in remaining bubbles with your best guess (or even a single letter — it&apos;s often a B or C).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Use the flag-for-review feature&lt;/strong&gt; on questions you&apos;re unsure about. You can come back to them within the same module.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Track time per question.&lt;/strong&gt; R&amp;#x26;W gives you ~71 sec; Math gives you ~95 sec. If a question is taking 3× that, flag it and move on.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&quot;test-day-timeline-full-version&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#test-day-timeline-full-version&quot; aria-label=&quot;test day timeline full version permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Test day timeline (full version)&lt;/h2&gt;
&lt;p&gt;If your test starts at 8:00 AM, here&apos;s the realistic timeline:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;6:30 AM&lt;/strong&gt; — wake up, eat protein-heavy breakfast&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;7:15 AM&lt;/strong&gt; — leave for test center&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;7:45 AM&lt;/strong&gt; — arrive (15-min buffer for traffic, parking, finding the room)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;8:00 AM&lt;/strong&gt; — check-in begins&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;8:35 AM&lt;/strong&gt; — instructions, Bluebook unlocks&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;8:45 AM&lt;/strong&gt; — Reading &amp;#x26; Writing Module 1 starts&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;9:17 AM&lt;/strong&gt; — Module 1 ends, Module 2 starts immediately&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;9:49 AM&lt;/strong&gt; — R&amp;#x26;W Module 2 ends, &lt;strong&gt;break begins&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;9:59 AM&lt;/strong&gt; — break ends, Math Module 1 starts&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;10:34 AM&lt;/strong&gt; — Math Module 1 ends, Math Module 2 starts immediately&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;11:09 AM&lt;/strong&gt; — Math Module 2 ends, dismissal&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;11:20 AM&lt;/strong&gt; — out the door&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Total time committed: ~5 hours including travel.&lt;/strong&gt;&lt;/p&gt;
&lt;h2 id=&quot;what-to-bring-to-make-timing-work&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#what-to-bring-to-make-timing-work&quot; aria-label=&quot;what to bring to make timing work permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What to bring to make timing work&lt;/h2&gt;
&lt;p&gt;The right materials let the testing time be the only time you spend stressed:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Charged device&lt;/strong&gt; — Bluebook runs on Mac, Windows, iPad, Chromebook&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Charger&lt;/strong&gt; — outlets aren&apos;t guaranteed; bring it anyway&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Photo ID&lt;/strong&gt; — driver&apos;s license, passport, or school ID with photo&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Admission ticket&lt;/strong&gt; — printed + screenshot as backup&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Approved physical calculator&lt;/strong&gt; — Desmos is on-screen, but a TI-84 backup never hurts&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Snack + water bottle&lt;/strong&gt; for the break&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Sweatshirt&lt;/strong&gt; — test rooms are usually cold&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&quot;faqs&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#faqs&quot; aria-label=&quot;faqs permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;FAQs&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;How long is the Digital SAT in total?&lt;/strong&gt;
2 hours 14 minutes of testing time + 10 minutes break = 2:24 of locked-in test time. With check-in and dismissal, plan for ~3 hours at the center.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is the Digital SAT shorter than the paper SAT?&lt;/strong&gt;
Yes — 46 minutes shorter. The paper SAT was 3:00 of testing; the Digital SAT is 2:14. Same scoring scale, same admissions weight.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Can I take a break during the SAT?&lt;/strong&gt;
You get one 10-minute break, between R&amp;#x26;W and Math. You can&apos;t take an unscheduled break without forfeiting test time (and possibly being marked as having left the test).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Do I get extra time on the Digital SAT?&lt;/strong&gt;
Only if you&apos;ve been approved for accommodations through College Board&apos;s SSD program. Apply at least 7 weeks before your target test date through your school&apos;s SSD coordinator.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Why is the SAT so much shorter now?&lt;/strong&gt;
The adaptive testing model means College Board can measure your skill level with fewer questions. Module 1&apos;s results route you to a Module 2 calibrated to your level, which gives more scoring precision per minute than the paper SAT&apos;s fixed-difficulty approach.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is the test the same length internationally?&lt;/strong&gt;
Yes — the Digital SAT is identical worldwide. Same 2:14 of testing time, same number of questions, same break structure.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;How long does it take to study for the SAT?&lt;/strong&gt;
Most students need 8–12 weeks of consistent study to move their score meaningfully. See our &lt;a href=&quot;/blog/sat-test-dates-2026/&quot;&gt;SAT test dates 2026 guide&lt;/a&gt; for picking a target date that gives you that runway.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;What&apos;s a good Digital SAT score?&lt;/strong&gt;
Anything above 1050 is above the national average. See our &lt;a href=&quot;/blog/good-sat-score/&quot;&gt;good SAT score guide&lt;/a&gt; for college-by-college targets.&lt;/p&gt;</content:encoded></item><item><title><![CDATA[Best SAT Question Banks Compared (2026): Free, Paid & AI-Powered]]></title><description><![CDATA[We compared the major SAT question banks — College Board, Khan Academy, Princeton Review, Kaplan, and Makon AI — across question count, adaptive accuracy, explanations, and price. Here's what's actually worth using in 2026.]]></description><link>https://www.makon.ai/blog/best-sat-question-banks</link><guid isPermaLink="false">https://www.makon.ai/blog/best-sat-question-banks</guid><pubDate>Mon, 27 Apr 2026 00:00:00 GMT</pubDate><content:encoded>&lt;p&gt;The single best predictor of SAT score growth isn&apos;t hours studied — it&apos;s the quality of the questions you study with. A great SAT question bank gives you exam-realistic problems, explains every wrong answer, and adapts as you improve. A bad one drills you on outdated content from the paper SAT and burns weeks of prep time.&lt;/p&gt;
&lt;p&gt;This guide compares the five major SAT question banks students actually use in 2026, ranked across the metrics that matter: question count, format match to the Digital SAT, explanation quality, adaptive routing, mobile experience, and price.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;TL;DR:&lt;/strong&gt; for free questions, College Board&apos;s official Bluebook practice tests + Khan Academy&apos;s question bank are the unbeatable starting point. For deeper drills with AI-graded feedback, &lt;a href=&quot;https://app.makon.ai&quot;&gt;Makon AI&lt;/a&gt; and Kaplan are the leaders. Princeton Review&apos;s bank is best for students who want a structured course wrapped around the questions.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;what-makes-a-good-sat-question-bank&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#what-makes-a-good-sat-question-bank&quot; aria-label=&quot;what makes a good sat question bank permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What makes a good SAT question bank?&lt;/h2&gt;
&lt;p&gt;Before the rankings — what should you actually look for?&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Digital SAT format match.&lt;/strong&gt; Many older question banks still use legacy paper SAT content. The Digital SAT changed reading comprehension (single-passage now), removed the no-calculator math section, and added section-adaptive scoring. If the bank&apos;s questions don&apos;t match these formats, you&apos;re prepping for the wrong test.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Question count.&lt;/strong&gt; A small bank means repetition by week 4. Aim for 1,500+ questions across both sections, ideally 3,000+.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Explanation quality.&lt;/strong&gt; &quot;The correct answer is C&quot; isn&apos;t an explanation. Look for banks that walk you through &lt;em&gt;why&lt;/em&gt; C is right and &lt;em&gt;why&lt;/em&gt; the other options are tempting traps.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Adaptive difficulty.&lt;/strong&gt; A great bank gets harder as you get better. Otherwise you spend weeks on questions you already know.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Performance tracking.&lt;/strong&gt; You can&apos;t improve what you can&apos;t see. The bank should show your accuracy by content area (algebra, advanced math, R&amp;#x26;W information vs. ideas, etc.) and trend it over time.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Price.&lt;/strong&gt; Free is great, but price-per-question and price-per-explanation matter more than the absolute monthly cost.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;With those criteria in mind, here are the rankings.&lt;/p&gt;
&lt;h2 id=&quot;1-college-board-bluebook-free--official-practice-tests&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#1-college-board-bluebook-free--official-practice-tests&quot; aria-label=&quot;1 college board bluebook free  official practice tests permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;1. College Board Bluebook (free) — official practice tests&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Metric&lt;/th&gt;
&lt;th&gt;Score&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Question count&lt;/td&gt;
&lt;td&gt;~600 (across 6 full-length practice tests + the Question Bank)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Digital SAT format match&lt;/td&gt;
&lt;td&gt;✅ Perfect — these &lt;em&gt;are&lt;/em&gt; official practice tests&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Explanations&lt;/td&gt;
&lt;td&gt;❌ Minimal — you get the correct answer, no rationale&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Adaptive&lt;/td&gt;
&lt;td&gt;✅ Real adaptive routing on full-length tests&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Performance tracking&lt;/td&gt;
&lt;td&gt;✅ Section-level scores, no item-level analytics&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Price&lt;/td&gt;
&lt;td&gt;Free&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;&lt;strong&gt;What it&apos;s good for:&lt;/strong&gt; there is no substitute for College Board&apos;s own questions. The 6 official Bluebook practice tests are calibrated to the real exam by the company that writes the real exam. Take all of them — at minimum, take them as your &lt;strong&gt;first&lt;/strong&gt; and &lt;strong&gt;last&lt;/strong&gt; practice tests in any prep cycle.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Where it falls short:&lt;/strong&gt; Bluebook&apos;s question bank doesn&apos;t explain wrong answers. You see &quot;you got Q14 wrong, the correct answer was B&quot; and that&apos;s it. For students who already know the content well enough to teach themselves, that&apos;s fine; for everyone else, you&apos;ll need a second tool that does explain.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Best for:&lt;/strong&gt; every student. Use it as your foundation, not your full study tool.&lt;/p&gt;
&lt;h2 id=&quot;2-khan-academy-official-sat-practice-free&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#2-khan-academy-official-sat-practice-free&quot; aria-label=&quot;2 khan academy official sat practice free permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;2. Khan Academy Official SAT Practice (free)&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Metric&lt;/th&gt;
&lt;th&gt;Score&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Question count&lt;/td&gt;
&lt;td&gt;~3,500 (curated by College Board)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Digital SAT format match&lt;/td&gt;
&lt;td&gt;✅ Updated for the Digital SAT in 2024&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Explanations&lt;/td&gt;
&lt;td&gt;✅ Step-by-step video walkthroughs&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Adaptive&lt;/td&gt;
&lt;td&gt;⚠️ Light — recommends questions based on practice test results&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Performance tracking&lt;/td&gt;
&lt;td&gt;✅ Strong — content-area heatmap&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Price&lt;/td&gt;
&lt;td&gt;Free&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;&lt;strong&gt;What it&apos;s good for:&lt;/strong&gt; the explanation depth. Every question has a video walkthrough showing the worked solution and common wrong-answer logic. The &quot;level up&quot; recommendations adjust difficulty based on your last few sessions, so you&apos;re not stuck on questions that are too easy.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Where it falls short:&lt;/strong&gt; the adaptive logic isn&apos;t true module-level routing. It recommends sets of questions but doesn&apos;t replicate the Digital SAT&apos;s harder/easier Module 2 split. You&apos;ll be technically prepared but won&apos;t have practiced the routing dynamic itself.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Best for:&lt;/strong&gt; students aiming 1100–1450 who need broad content coverage with patient explanations. The free price is genuinely unbeatable.&lt;/p&gt;
&lt;h2 id=&quot;3-makon-ai-question-bank-free-tier--paid-plans&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#3-makon-ai-question-bank-free-tier--paid-plans&quot; aria-label=&quot;3 makon ai question bank free tier  paid plans permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;3. Makon AI Question Bank (free tier + paid plans)&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Metric&lt;/th&gt;
&lt;th&gt;Score&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Question count&lt;/td&gt;
&lt;td&gt;5,000+ Digital SAT-format questions&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Digital SAT format match&lt;/td&gt;
&lt;td&gt;✅ Native Digital SAT, single-passage R&amp;#x26;W&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Explanations&lt;/td&gt;
&lt;td&gt;✅ AI tutor (Po) explains every miss + suggests follow-up drills&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Adaptive&lt;/td&gt;
&lt;td&gt;✅ Module-by-module adaptive testing + targeted weak-area drills&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Performance tracking&lt;/td&gt;
&lt;td&gt;✅ Per-skill granularity, error logging built in&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Price&lt;/td&gt;
&lt;td&gt;Free trial + paid tiers&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;&lt;strong&gt;What it&apos;s good for:&lt;/strong&gt; the AI tutor that walks you through every wrong answer and then &lt;strong&gt;builds a custom drill&lt;/strong&gt; based on what you just missed. That feedback loop — miss → explanation → targeted retry — is the highest-leverage feature in modern SAT prep. Most banks stop at the explanation; Makon AI turns the explanation into your next practice set.&lt;/p&gt;
&lt;p&gt;The question bank is also the largest in this list at 5,000+ Digital SAT-format questions, all calibrated to mirror real exam difficulty.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Where it falls short:&lt;/strong&gt; unlike Khan Academy, Makon AI&apos;s deepest features sit behind paid tiers after the free trial. The free tier is enough to evaluate fit; serious test prep eventually steps up.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Best for:&lt;/strong&gt; students 1200+ who want adaptive depth and an always-on AI tutor. Try it free at &lt;a href=&quot;https://app.makon.ai&quot;&gt;app.makon.ai&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id=&quot;4-kaplan-sat-prep-200700-depending-on-plan&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#4-kaplan-sat-prep-200700-depending-on-plan&quot; aria-label=&quot;4 kaplan sat prep 200700 depending on plan permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;4. Kaplan SAT Prep ($200–$700 depending on plan)&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Metric&lt;/th&gt;
&lt;th&gt;Score&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Question count&lt;/td&gt;
&lt;td&gt;~2,000 + 8 full-length practice tests&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Digital SAT format match&lt;/td&gt;
&lt;td&gt;✅ Updated for the Digital SAT&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Explanations&lt;/td&gt;
&lt;td&gt;✅ Strong, traditional textbook-style&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Adaptive&lt;/td&gt;
&lt;td&gt;⚠️ Practice test scoring is adaptive; question bank itself isn&apos;t&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Performance tracking&lt;/td&gt;
&lt;td&gt;✅ Smart Reports show pacing + accuracy&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Price&lt;/td&gt;
&lt;td&gt;$200 (self-paced) – $700 (live online course)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;&lt;strong&gt;What it&apos;s good for:&lt;/strong&gt; Kaplan&apos;s content is rock-solid. Their explanations read like a polished textbook — you understand the rule, see the worked example, then drill it. Strategy chapters between question sets give you frameworks (e.g., when to plug in numbers, when to back-solve) that other banks skip entirely.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Where it falls short:&lt;/strong&gt; at $200+ for self-paced and $700+ for live courses, Kaplan is one of the priciest options. The mobile experience trails the leaders — most students end up using Kaplan on a laptop, not on the phone during commutes.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Best for:&lt;/strong&gt; students who learn from structured curricula and have the budget. Go for the full course, not just the question bank — the strategy content is where Kaplan earns its price.&lt;/p&gt;
&lt;h2 id=&quot;5-princeton-review-sat-question-bank-1001400&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#5-princeton-review-sat-question-bank-1001400&quot; aria-label=&quot;5 princeton review sat question bank 1001400 permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;5. Princeton Review SAT Question Bank ($100–$1,400)&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Metric&lt;/th&gt;
&lt;th&gt;Score&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Question count&lt;/td&gt;
&lt;td&gt;~2,500 + 10 full-length practice tests (Premium)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Digital SAT format match&lt;/td&gt;
&lt;td&gt;✅ Updated for the Digital SAT&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Explanations&lt;/td&gt;
&lt;td&gt;✅ Strong, slightly more conversational than Kaplan&apos;s&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Adaptive&lt;/td&gt;
&lt;td&gt;⚠️ Practice tests adaptive, drill mode static&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Performance tracking&lt;/td&gt;
&lt;td&gt;✅ Score reports with per-section + per-question-type breakdowns&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Price&lt;/td&gt;
&lt;td&gt;$100 (Self-Paced) – $1,400 (Tutor + Course bundles)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;&lt;strong&gt;What it&apos;s good for:&lt;/strong&gt; the &lt;strong&gt;dual prep + question bank&lt;/strong&gt; structure. You&apos;re not just answering questions — you&apos;re working through a structured curriculum with built-in drills, then reinforcing with full-length practice tests. The &quot;Score Improvement Guarantee&quot; (your money back if you don&apos;t beat your starting score) is a real differentiator at the higher tiers.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Where it falls short:&lt;/strong&gt; the cheapest tier ($100) is mostly content review, not active drilling. To get the meaningful question bank, you need the $400+ tier. The mobile app is functional but feels dated relative to Khan Academy or Makon AI.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Best for:&lt;/strong&gt; students who want a &quot;complete program&quot; with accountability and aren&apos;t price-sensitive.&lt;/p&gt;
&lt;h2 id=&quot;honorable-mentions&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#honorable-mentions&quot; aria-label=&quot;honorable mentions permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Honorable mentions&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;The College Panda SAT Math&lt;/strong&gt; — book-only, but the math problem set is widely praised by 1500+ scorers. Pair with any digital bank.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Bluebook Daily Practice&lt;/strong&gt; — College Board&apos;s free daily question, delivered via the Bluebook app. Tiny but adds up over months.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&quot;side-by-side-comparison&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#side-by-side-comparison&quot; aria-label=&quot;side by side comparison permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Side-by-side comparison&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Bank&lt;/th&gt;
&lt;th&gt;Questions&lt;/th&gt;
&lt;th&gt;Adaptive&lt;/th&gt;
&lt;th&gt;Explanations&lt;/th&gt;
&lt;th&gt;Price&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;College Board Bluebook&lt;/td&gt;
&lt;td&gt;~600&lt;/td&gt;
&lt;td&gt;✅ True&lt;/td&gt;
&lt;td&gt;❌ None&lt;/td&gt;
&lt;td&gt;Free&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Khan Academy&lt;/td&gt;
&lt;td&gt;3,500&lt;/td&gt;
&lt;td&gt;⚠️ Light&lt;/td&gt;
&lt;td&gt;✅ Video&lt;/td&gt;
&lt;td&gt;Free&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Makon AI&lt;/td&gt;
&lt;td&gt;5,000+&lt;/td&gt;
&lt;td&gt;✅ True&lt;/td&gt;
&lt;td&gt;✅ AI tutor&lt;/td&gt;
&lt;td&gt;Free trial + paid&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Kaplan&lt;/td&gt;
&lt;td&gt;2,000&lt;/td&gt;
&lt;td&gt;⚠️ Practice tests only&lt;/td&gt;
&lt;td&gt;✅ Textbook&lt;/td&gt;
&lt;td&gt;$200–$700&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Princeton Review&lt;/td&gt;
&lt;td&gt;2,500&lt;/td&gt;
&lt;td&gt;⚠️ Practice tests only&lt;/td&gt;
&lt;td&gt;✅ Conversational&lt;/td&gt;
&lt;td&gt;$100–$1,400&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;h2 id=&quot;how-to-actually-use-these-together&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-to-actually-use-these-together&quot; aria-label=&quot;how to actually use these together permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How to actually use these together&lt;/h2&gt;
&lt;p&gt;Don&apos;t pick just one. The strongest prep stacks combine 2–3:&lt;/p&gt;
&lt;h3 id=&quot;the-free-stack-best-for-budget-conscious&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-free-stack-best-for-budget-conscious&quot; aria-label=&quot;the free stack best for budget conscious permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The free stack (best for budget-conscious)&lt;/h3&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;College Board Bluebook&lt;/strong&gt; — take 1 official practice test up front for your baseline, save 5 more for the back half of prep.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Khan Academy&lt;/strong&gt; — your daily question bank for content review and weak-area drilling.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Daily Bluebook practice question&lt;/strong&gt; — 1 free question per day to keep the format fresh.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;This combination gets most students from baseline to ~1300 without spending a dollar.&lt;/p&gt;
&lt;h3 id=&quot;the-accelerated-stack-best-for-1300--1500&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-accelerated-stack-best-for-1300--1500&quot; aria-label=&quot;the accelerated stack best for 1300  1500 permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The accelerated stack (best for 1300+ → 1500)&lt;/h3&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;College Board Bluebook&lt;/strong&gt; for benchmarks.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Makon AI&lt;/strong&gt; for adaptive drills + AI explanations.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;The College Panda SAT Math&lt;/strong&gt; book if math is your weak section.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;The AI feedback loop in Makon AI is the meaningful upgrade over Khan once you&apos;re past the basics. Khan Academy&apos;s video explanations are excellent at the content-review level; Makon AI&apos;s per-error follow-ups become more useful when you need targeted weak-spot work.&lt;/p&gt;
&lt;h3 id=&quot;the-structured-stack-best-for-self-pacers-who-need-accountability&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#the-structured-stack-best-for-self-pacers-who-need-accountability&quot; aria-label=&quot;the structured stack best for self pacers who need accountability permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;The structured stack (best for self-pacers who need accountability)&lt;/h3&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Princeton Review SAT Premium&lt;/strong&gt; ($400) for the wraparound curriculum.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;College Board Bluebook&lt;/strong&gt; for the official benchmarks.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Khan Academy&lt;/strong&gt; as a free supplement for math content review.&lt;/li&gt;
&lt;/ol&gt;
&lt;h2 id=&quot;common-mistakes-when-picking-a-question-bank&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#common-mistakes-when-picking-a-question-bank&quot; aria-label=&quot;common mistakes when picking a question bank permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Common mistakes when picking a question bank&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Buying everything.&lt;/strong&gt; Three banks &gt; five. More question banks means more switching cost, more half-finished progress trackers, and worse focus.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Choosing based on question count alone.&lt;/strong&gt; A bank with 5,000 great-but-easy questions is worse than 1,500 well-calibrated harder ones. Match difficulty to your current score.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Skipping official practice tests.&lt;/strong&gt; No matter what bank you use day-to-day, anchor your prep with all 6 College Board Bluebook tests. Anything else is an approximation.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Ignoring mobile.&lt;/strong&gt; You&apos;ll do 30%+ of your prep on the phone during commutes and 5-min downtime. If the mobile app feels clunky, you&apos;ll skip those sessions and they&apos;re free score points.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Not tracking your error log.&lt;/strong&gt; The bank tracks accuracy. &lt;em&gt;You&lt;/em&gt; should track &lt;em&gt;why&lt;/em&gt; you got it wrong: misread the question, careless arithmetic, didn&apos;t know the content, ran out of time. Pattern recognition on your own errors is the highest-leverage skill in SAT prep.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&quot;faqs&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#faqs&quot; aria-label=&quot;faqs permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;FAQs&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;What&apos;s the best free SAT question bank?&lt;/strong&gt;
Khan Academy Official SAT Practice. ~3,500 curated College Board questions, free, with video explanations on every problem.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Are official College Board practice tests adaptive?&lt;/strong&gt;
The full-length practice tests inside Bluebook are. The standalone Question Bank delivers fixed-difficulty questions.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;How many SAT questions should I do in total prep?&lt;/strong&gt;
A typical 8–12 week prep plan covers 800–1,500 questions. Quality &gt; quantity — every question you do should produce a learning, not just an answer.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;What&apos;s the difference between adaptive testing and adaptive drilling?&lt;/strong&gt;
Adaptive &lt;strong&gt;testing&lt;/strong&gt; changes the difficulty of &lt;em&gt;upcoming&lt;/em&gt; questions based on past answers (mimicking the real Digital SAT). Adaptive &lt;strong&gt;drilling&lt;/strong&gt; picks practice problems based on which content areas you&apos;re weakest in. Most banks do one or the other; Makon AI and Kaplan do both.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Should I use Khan Academy or Makon AI?&lt;/strong&gt;
If you&apos;re under 1300 and budget-constrained: Khan Academy. If you&apos;re 1200+ and willing to pay for AI-driven personalization: try Makon AI&apos;s free trial.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Are Princeton Review and Kaplan still worth it in the AI era?&lt;/strong&gt;
Yes, for the wraparound curriculum. They&apos;re not winning on raw question count or AI features, but their structured courses and live tutoring (in the higher tiers) give some students the accountability they need.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;How long should I spend on each question?&lt;/strong&gt;
Roughly 71 seconds per R&amp;#x26;W question and 95 seconds per Math question — this matches the &lt;a href=&quot;/blog/how-long-is-the-sat/&quot;&gt;actual SAT pacing&lt;/a&gt;. If you&apos;re consistently 2x slower, you have a content gap; if you&apos;re slower only in Math Module 2, it&apos;s a pacing problem.&lt;/p&gt;
&lt;h2 id=&quot;pick-one-and-start&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#pick-one-and-start&quot; aria-label=&quot;pick one and start permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Pick one and start&lt;/h2&gt;
&lt;p&gt;The biggest mistake students make isn&apos;t picking the &lt;em&gt;wrong&lt;/em&gt; question bank — it&apos;s spending two weeks researching and not starting. Every bank in this list will move your score if you actually use it consistently. Pick one, commit to 5 sessions a week, and revisit the question after a month.&lt;/p&gt;
&lt;p&gt;If you want to start with a 5-minute test of Makon AI&apos;s question bank + AI tutor, &lt;a href=&quot;https://app.makon.ai&quot;&gt;the free trial is here&lt;/a&gt;. Otherwise, open Bluebook and take your first official practice test today — that&apos;s the baseline every prep plan needs.&lt;/p&gt;</content:encoded></item><item><title><![CDATA[What’s a Good SAT Score in 2026? (And What Your Dream School Actually Expects)]]></title><description><![CDATA[A good SAT score depends on your target colleges. Here’s the average SAT score, the percentile breakdown, and the score ranges that get you into Ivy League, top public, and state schools.]]></description><link>https://www.makon.ai/blog/good-sat-score</link><guid isPermaLink="false">https://www.makon.ai/blog/good-sat-score</guid><pubDate>Mon, 20 Apr 2026 00:00:00 GMT</pubDate><content:encoded>&lt;p&gt;“Is &lt;strong&gt;1300&lt;/strong&gt; a good SAT score?” “What about &lt;strong&gt;1450&lt;/strong&gt;?” The honest answer: it depends on the colleges you’re applying to. A 1300 is a 90th-percentile score that gets you into great state schools but is below the 25th percentile at MIT. This guide gives you a clean way to read your score, the official 2026 score ranges, and the score targets you should aim at for the schools you actually care about.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Skip ahead:&lt;/strong&gt; plug your raw correct answers into our &lt;a href=&quot;/sat-score-calculator&quot;&gt;free Digital SAT score calculator&lt;/a&gt; to see where your latest practice test lands you on these tiers.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;what-is-a-good-sat-score&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#what-is-a-good-sat-score&quot; aria-label=&quot;what is a good sat score permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What is a good SAT score?&lt;/h2&gt;
&lt;p&gt;The simplest answer: any score &lt;strong&gt;above the 50th percentile (~1050)&lt;/strong&gt; is above-average. But that bar is meaningless for most students applying to four-year colleges. A more useful framing:&lt;/p&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Total score&lt;/th&gt;
&lt;th&gt;Percentile&lt;/th&gt;
&lt;th&gt;What it means&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;1600&lt;/td&gt;
&lt;td&gt;99.95th&lt;/td&gt;
&lt;td&gt;Perfect score (≈300 students/year nationally)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;1500+&lt;/td&gt;
&lt;td&gt;99th&lt;/td&gt;
&lt;td&gt;Top 1% — competitive at every Ivy League school&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;1400–1499&lt;/td&gt;
&lt;td&gt;95th&lt;/td&gt;
&lt;td&gt;Strong for selective colleges (top 30)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;1300–1399&lt;/td&gt;
&lt;td&gt;87th&lt;/td&gt;
&lt;td&gt;Competitive for most four-year universities&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;1200–1299&lt;/td&gt;
&lt;td&gt;75th&lt;/td&gt;
&lt;td&gt;Solid; competitive at most state schools&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;1100–1199&lt;/td&gt;
&lt;td&gt;60th&lt;/td&gt;
&lt;td&gt;Above average; many solid options&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;1000–1099&lt;/td&gt;
&lt;td&gt;41st&lt;/td&gt;
&lt;td&gt;Around the &lt;strong&gt;average SAT score&lt;/strong&gt; (national mean ≈ 1050)&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;900–999&lt;/td&gt;
&lt;td&gt;25th&lt;/td&gt;
&lt;td&gt;Below average; community college and open-admission programs&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;400–899&lt;/td&gt;
&lt;td&gt;bottom 25%&lt;/td&gt;
&lt;td&gt;Lowest tier; retake recommended&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p&gt;Percentiles are rounded based on College Board’s 2025 SAT Suite of Assessments report (the most recent national distribution).&lt;/p&gt;
&lt;h2 id=&quot;what-is-the-average-sat-score&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#what-is-the-average-sat-score&quot; aria-label=&quot;what is the average sat score permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What is the average SAT score?&lt;/h2&gt;
&lt;p&gt;The &lt;strong&gt;national average SAT score&lt;/strong&gt; is approximately &lt;strong&gt;1050&lt;/strong&gt; (based on the latest College Board data for graduating seniors in 2025). That breaks down to:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Reading &amp;#x26; Writing average:&lt;/strong&gt; ~525&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Math average:&lt;/strong&gt; ~520&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;The average has hovered between 1030 and 1080 over the last decade. It tends to drift down slightly each year because more students take the test.&lt;/p&gt;
&lt;h2 id=&quot;whats-a-good-sat-score-for-college-admissions&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#whats-a-good-sat-score-for-college-admissions&quot; aria-label=&quot;whats a good sat score for college admissions permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What’s a good SAT score for college admissions?&lt;/h2&gt;
&lt;p&gt;Here’s where you actually need to plant your flag. Below are the &lt;strong&gt;middle-50% SAT score ranges&lt;/strong&gt; for admitted students at popular schools (the 25th percentile to the 75th percentile of admitted students). Your goal: aim for the &lt;strong&gt;75th-percentile&lt;/strong&gt; number to be a competitive applicant.&lt;/p&gt;
&lt;h3 id=&quot;ivy-league--ivy-1450--1580-range&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#ivy-league--ivy-1450--1580-range&quot; aria-label=&quot;ivy league  ivy 1450  1580 range permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Ivy League &amp;#x26; Ivy+ (1450 – 1580 range)&lt;/h3&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;School&lt;/th&gt;
&lt;th&gt;25th – 75th %&lt;/th&gt;
&lt;th&gt;Aim for&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Harvard University&lt;/td&gt;
&lt;td&gt;1500 – 1580&lt;/td&gt;
&lt;td&gt;1550+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Princeton University&lt;/td&gt;
&lt;td&gt;1500 – 1580&lt;/td&gt;
&lt;td&gt;1550+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;MIT&lt;/td&gt;
&lt;td&gt;1530 – 1580&lt;/td&gt;
&lt;td&gt;1570+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Yale University&lt;/td&gt;
&lt;td&gt;1500 – 1580&lt;/td&gt;
&lt;td&gt;1550+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Stanford University&lt;/td&gt;
&lt;td&gt;1500 – 1580&lt;/td&gt;
&lt;td&gt;1550+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Columbia University&lt;/td&gt;
&lt;td&gt;1490 – 1570&lt;/td&gt;
&lt;td&gt;1540+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;University of Chicago&lt;/td&gt;
&lt;td&gt;1510 – 1580&lt;/td&gt;
&lt;td&gt;1550+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Caltech&lt;/td&gt;
&lt;td&gt;1530 – 1580&lt;/td&gt;
&lt;td&gt;1570+&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;h3 id=&quot;top-public--private-1380--1530-range&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#top-public--private-1380--1530-range&quot; aria-label=&quot;top public  private 1380  1530 range permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Top public + private (1380 – 1530 range)&lt;/h3&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;School&lt;/th&gt;
&lt;th&gt;25th – 75th %&lt;/th&gt;
&lt;th&gt;Aim for&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;UC Berkeley&lt;/td&gt;
&lt;td&gt;1330 – 1530&lt;/td&gt;
&lt;td&gt;1500+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;UCLA&lt;/td&gt;
&lt;td&gt;1290 – 1510&lt;/td&gt;
&lt;td&gt;1480+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;University of Michigan&lt;/td&gt;
&lt;td&gt;1350 – 1530&lt;/td&gt;
&lt;td&gt;1500+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;University of Virginia&lt;/td&gt;
&lt;td&gt;1380 – 1500&lt;/td&gt;
&lt;td&gt;1480+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Georgia Tech&lt;/td&gt;
&lt;td&gt;1370 – 1510&lt;/td&gt;
&lt;td&gt;1480+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Carnegie Mellon&lt;/td&gt;
&lt;td&gt;1480 – 1560&lt;/td&gt;
&lt;td&gt;1530+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;NYU&lt;/td&gt;
&lt;td&gt;1370 – 1530&lt;/td&gt;
&lt;td&gt;1500+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Boston University&lt;/td&gt;
&lt;td&gt;1330 – 1500&lt;/td&gt;
&lt;td&gt;1470+&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;h3 id=&quot;strong-state-flagship-1200--1400-range&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#strong-state-flagship-1200--1400-range&quot; aria-label=&quot;strong state flagship 1200  1400 range permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Strong state flagship (1200 – 1400 range)&lt;/h3&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;School&lt;/th&gt;
&lt;th&gt;25th – 75th %&lt;/th&gt;
&lt;th&gt;Aim for&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;University of Texas Austin&lt;/td&gt;
&lt;td&gt;1230 – 1480&lt;/td&gt;
&lt;td&gt;1450+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Penn State&lt;/td&gt;
&lt;td&gt;1210 – 1410&lt;/td&gt;
&lt;td&gt;1380+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;University of Florida&lt;/td&gt;
&lt;td&gt;1300 – 1450&lt;/td&gt;
&lt;td&gt;1430+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;University of Wisconsin-Madison&lt;/td&gt;
&lt;td&gt;1290 – 1460&lt;/td&gt;
&lt;td&gt;1430+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Ohio State&lt;/td&gt;
&lt;td&gt;1240 – 1420&lt;/td&gt;
&lt;td&gt;1400+&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;University of Washington&lt;/td&gt;
&lt;td&gt;1210 – 1440&lt;/td&gt;
&lt;td&gt;1410+&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;h3 id=&quot;solid-four-year-1050--1200-range&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#solid-four-year-1050--1200-range&quot; aria-label=&quot;solid four year 1050  1200 range permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Solid four-year (1050 – 1200 range)&lt;/h3&gt;
&lt;p&gt;Most regional state universities (e.g., Arizona State, University of Arizona, San Diego State, University of Iowa) admit students with SAT scores from roughly &lt;strong&gt;1050 to 1230&lt;/strong&gt;. A 1200 puts you above the 75th percentile at many of these.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Score ranges shift year to year.&lt;/strong&gt; Always check each college’s most recent Common Data Set (Section C9) for current numbers. The figures above are from 2024–2025 admit cycles.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;whats-a-high-sat-score&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#whats-a-high-sat-score&quot; aria-label=&quot;whats a high sat score permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What’s a high SAT score?&lt;/h2&gt;
&lt;p&gt;“High” means different things in different contexts. Here’s a reasonable framing:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;High for the average student:&lt;/strong&gt; 1300+ (top 13%)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;High for selective admissions:&lt;/strong&gt; 1450+ (top 5%)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;High for top-30 schools:&lt;/strong&gt; 1500+ (top 1%)&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;High for top-10 schools:&lt;/strong&gt; 1550+ (top 0.5%)&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;A 1500 isn’t the same as a 1550 to elite admissions, but past 1500 the marginal points matter less than your essays, GPA, and extracurriculars.&lt;/p&gt;
&lt;h2 id=&quot;whats-a-perfect-sat-score&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#whats-a-perfect-sat-score&quot; aria-label=&quot;whats a perfect sat score permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What’s a perfect SAT score?&lt;/h2&gt;
&lt;p&gt;A &lt;strong&gt;perfect SAT score is 1600&lt;/strong&gt; — 800 in Reading &amp;#x26; Writing + 800 in Math. About &lt;strong&gt;300 to 400 students per year&lt;/strong&gt; earn it, out of nearly 2 million test takers. That’s 0.02%.&lt;/p&gt;
&lt;p&gt;Worth noting: a 1580 or 1590 is &lt;strong&gt;functionally equivalent&lt;/strong&gt; to a 1600 for admissions. Most colleges superscore (combine your highest section scores across multiple sittings), so even 790 + 790 effectively becomes 1600 if you nail each section on different test dates.&lt;/p&gt;
&lt;h2 id=&quot;section-scores-whats-a-good-score-in-each&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#section-scores-whats-a-good-score-in-each&quot; aria-label=&quot;section scores whats a good score in each permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Section scores: what’s a good score in each?&lt;/h2&gt;
&lt;p&gt;The 1600 total is split equally between two sections, each scored 200–800. Section averages and benchmarks:&lt;/p&gt;
&lt;h3 id=&quot;reading--writing&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#reading--writing&quot; aria-label=&quot;reading  writing permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Reading &amp;#x26; Writing&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;Average: 525&lt;/li&gt;
&lt;li&gt;Good: 650+&lt;/li&gt;
&lt;li&gt;Strong (top 10%): 720+&lt;/li&gt;
&lt;li&gt;Elite: 770+&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&quot;math&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#math&quot; aria-label=&quot;math permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Math&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;Average: 520&lt;/li&gt;
&lt;li&gt;Good: 650+&lt;/li&gt;
&lt;li&gt;Strong (top 10%): 720+&lt;/li&gt;
&lt;li&gt;Elite: 770+&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;If your scores are uneven (say, 760 R&amp;#x26;W / 620 Math), prioritize closing the gap on the weaker side. Total scores rise faster from raising a 600 to 700 than from raising a 760 to 800.&lt;/p&gt;
&lt;h2 id=&quot;which-section-is-harder-to-improve&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#which-section-is-harder-to-improve&quot; aria-label=&quot;which section is harder to improve permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Which section is harder to improve?&lt;/h2&gt;
&lt;p&gt;Across the students we’ve worked with, &lt;strong&gt;Math is the easier section to move quickly&lt;/strong&gt; — but only up to a point. Math is finite: there are roughly 30 patterns that get reused across the test, and once you know all 30, you stop missing easy points. R&amp;#x26;W is harder to grind because every passage is new — you can’t memorize content the way you can memorize quadratic factoring.&lt;/p&gt;
&lt;p&gt;Practical rule of thumb:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;From 500 → 650&lt;/strong&gt; in Math: usually achievable in 6–8 weeks of focused practice. Most points come from cleaning up algebra and word-problem translation errors, not from learning new content.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;From 650 → 750&lt;/strong&gt; in Math: usually 10–12 weeks. Now you’re losing points to careless mistakes on hard problems, which require timed practice and pacing — not more content review.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;From 500 → 650&lt;/strong&gt; in R&amp;#x26;W: usually 8–10 weeks. The skill that grows fastest is reading the question stem precisely, not &quot;reading more comprehensively.&quot;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;From 650 → 750&lt;/strong&gt; in R&amp;#x26;W: usually 12+ weeks. At this band, you’re grinding for 2–3 hard inference / vocabulary-in-context questions per module, and growth slows considerably.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;If you’re hitting a hard ceiling in either section, the bottleneck is almost always pacing — not content gaps. Take a full-length practice test untimed and compare to your timed score. If untimed &gt; timed by 80+ points, your problem is time, not knowledge.&lt;/p&gt;
&lt;h2 id=&quot;score-improvement-tactics-by-score-band&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#score-improvement-tactics-by-score-band&quot; aria-label=&quot;score improvement tactics by score band permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Score-improvement tactics by score band&lt;/h2&gt;
&lt;p&gt;Different starting points need different strategies. The same advice doesn’t work at 1100 and 1500.&lt;/p&gt;
&lt;h3 id=&quot;1050--1300-250-point-lift&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#1050--1300-250-point-lift&quot; aria-label=&quot;1050  1300 250 point lift permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;1050 → 1300 (≈250-point lift)&lt;/h3&gt;
&lt;p&gt;This is the most common starting band, and the biggest jump per hour of study. Spend the first 4 weeks on &lt;strong&gt;content review&lt;/strong&gt; — algebra essentials, comma/semicolon grammar rules, transition words. The single highest-leverage thing is fixing systematic gaps. Most students at this band miss the same 5–6 topics over and over; identify them with a diagnostic and drill them until they’re cold.&lt;/p&gt;
&lt;h3 id=&quot;1300--1450-150-point-lift&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#1300--1450-150-point-lift&quot; aria-label=&quot;1300  1450 150 point lift permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;1300 → 1450 (≈150-point lift)&lt;/h3&gt;
&lt;p&gt;You already know the content. The bottleneck is &lt;strong&gt;accuracy under time pressure&lt;/strong&gt;. Switch to fully timed module-by-module practice. Track your error log: every wrong answer gets logged with &lt;em&gt;why&lt;/em&gt; you missed it (&quot;read the question wrong,&quot; &quot;didn’t check answer choices,&quot; &quot;ran out of time&quot;). Patterns emerge after 30–40 logged errors.&lt;/p&gt;
&lt;h3 id=&quot;1450--1550-100-point-lift&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#1450--1550-100-point-lift&quot; aria-label=&quot;1450  1550 100 point lift permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;1450 → 1550 (≈100-point lift)&lt;/h3&gt;
&lt;p&gt;You’re fighting for marginal points on the hardest 2–3 questions per module. Practice &lt;strong&gt;time discipline&lt;/strong&gt;: the goal is to finish each module with 4–5 minutes left, then come back and double-check your flagged questions. Section 2 of each module is the adaptive harder routing — make sure your Module 1 accuracy is at 95%+ to unlock the higher scoring ceiling.&lt;/p&gt;
&lt;h3 id=&quot;1550-perfectionist-territory&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#1550-perfectionist-territory&quot; aria-label=&quot;1550 perfectionist territory permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;1550+ (perfectionist territory)&lt;/h3&gt;
&lt;p&gt;Most of your missed points come from careless mistakes, not unknown content. The training data is your own error log. Read each missed problem and ask: &quot;What would have to be true about this question for me to answer it correctly?&quot; Then drill that exact pattern.&lt;/p&gt;
&lt;p&gt;Across all bands, the biggest predictor of score improvement isn’t hours studied — it’s whether you keep an &lt;strong&gt;error log&lt;/strong&gt;. Students who log every wrong answer typically improve 1.5× faster than students who just retake practice tests.&lt;/p&gt;
&lt;h2 id=&quot;whats-a-good-digital-sat-score-is-it-different-from-the-paper-sat&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#whats-a-good-digital-sat-score-is-it-different-from-the-paper-sat&quot; aria-label=&quot;whats a good digital sat score is it different from the paper sat permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What’s a good Digital SAT score? (Is it different from the paper SAT?)&lt;/h2&gt;
&lt;p&gt;The Digital SAT and the legacy paper SAT are scored on the &lt;strong&gt;same 400–1600 scale&lt;/strong&gt;, so the answer to “what’s a good score” doesn’t change. The score itself means the same thing.&lt;/p&gt;
&lt;p&gt;What did change with the digital format:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Adaptive routing.&lt;/strong&gt; Your Module 1 performance determines whether Module 2 routes you to the harder or easier set, which affects your scoring ceiling.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Shorter test.&lt;/strong&gt; 2:14 of testing time vs. 3:00 on the legacy SAT.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Fewer questions.&lt;/strong&gt; 98 total instead of 154.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Calculator on every Math question.&lt;/strong&gt; No more no-calculator section.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;College Board has confirmed they treat Digital SAT scores identically to legacy SAT scores for admissions.&lt;/p&gt;
&lt;h2 id=&quot;how-to-figure-out-what-you-need-to-score&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-to-figure-out-what-you-need-to-score&quot; aria-label=&quot;how to figure out what you need to score permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How to figure out what &lt;em&gt;you&lt;/em&gt; need to score&lt;/h2&gt;
&lt;p&gt;The cleanest 4-step method:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;List your top 5 college choices.&lt;/strong&gt; Yes, even if some are reaches.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;For each, look up the 75th-percentile SAT score&lt;/strong&gt; of admitted students (Common Data Set Section C9).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Take the highest of the five.&lt;/strong&gt; That’s your stretch goal.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Take a baseline SAT practice test today&lt;/strong&gt; to see your starting point. Plug raw correct answers into our &lt;a href=&quot;/sat-score-calculator&quot;&gt;Digital SAT score calculator&lt;/a&gt; for an instant scaled-score estimate.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;The gap between your baseline and your stretch goal is your study runway. Generally:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;0–50 points&lt;/strong&gt; to close: 4 weeks of focused review&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;50–150 points&lt;/strong&gt; to close: 8–10 weeks&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;150–300 points&lt;/strong&gt; to close: 12–16 weeks&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;300+ points&lt;/strong&gt;: rethink the goal or plan for two test sittings&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&quot;superscoring-vs-score-choice--what-your-score-actually-looks-like-to-colleges&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#superscoring-vs-score-choice--what-your-score-actually-looks-like-to-colleges&quot; aria-label=&quot;superscoring vs score choice  what your score actually looks like to colleges permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Superscoring vs Score Choice — what your score actually looks like to colleges&lt;/h2&gt;
&lt;p&gt;The total score on your transcript isn’t always what colleges see. Three different policies are in play depending on the school:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Superscoring (most schools).&lt;/strong&gt; They take your highest R&amp;#x26;W from any sitting and your highest Math from any sitting and combine them. So if you scored 720 R&amp;#x26;W / 620 Math in March and then 680 R&amp;#x26;W / 720 Math in October, your superscore is &lt;strong&gt;720 + 720 = 1440&lt;/strong&gt;, not 1400. This is why retaking is almost always worth it: you only need to improve one section per sitting.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Score Choice (College Board feature).&lt;/strong&gt; You pick which sittings get sent. If your June score was 1300 and your October score was 1450, you can send only October. Most schools allow this; a few require all sittings (notably Yale, Stanford, Penn, Georgetown, and military academies).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Single highest test (a small minority).&lt;/strong&gt; They take whichever single sitting was highest, with no superscoring. Boston College and Cornell historically use this.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;You don’t have to memorize which school does what — every college publishes its score-use policy on its admissions page or Common Data Set. The actionable takeaway: &lt;strong&gt;even if a school is &quot;all scores required,&quot; superscoring is the dominant policy at top-100 schools, so retaking is rarely a bad idea&lt;/strong&gt;.&lt;/p&gt;
&lt;h2 id=&quot;when-students-should-consider-the-act-instead&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#when-students-should-consider-the-act-instead&quot; aria-label=&quot;when students should consider the act instead permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;When students should consider the ACT instead&lt;/h2&gt;
&lt;p&gt;The ACT is scored on a 1–36 scale and tests slightly different content (it includes a Science section; Math has more geometry). Some students score visibly better on one than the other. A rough conversion:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;SAT 1200 ≈ ACT 25&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;SAT 1300 ≈ ACT 27&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;SAT 1400 ≈ ACT 31&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;SAT 1500 ≈ ACT 34&lt;/strong&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;If your SAT score is stuck at 1200 after a real effort and you haven’t tried the ACT, take an official ACT practice test. ~15% of students score noticeably higher on the ACT — usually because they’re strong on Science / data interpretation and weaker on the SAT’s reading inference questions. Colleges treat the two tests as fully equivalent, so you can submit whichever is higher.&lt;/p&gt;
&lt;p&gt;The reverse is also true: if you’re stuck on the ACT, the SAT may unlock a higher band. Don’t spend 6 months grinding on the wrong test.&lt;/p&gt;
&lt;h2 id=&quot;faqs&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#faqs&quot; aria-label=&quot;faqs permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;FAQs&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;Is 1200 a good SAT score?&lt;/strong&gt;
Yes — 1200 puts you above the 75th percentile of all test takers. It’s competitive at most state schools and many private colleges. For top-30 schools, you’ll want 1450+.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is 1400 a good SAT score?&lt;/strong&gt;
Yes — 1400 is a strong score (95th percentile). Competitive at most highly selective schools and a strong number for top-50 universities.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is 1500 a good SAT score?&lt;/strong&gt;
Yes — 1500 is in the top 1% nationally. Competitive at every Ivy League school, though most admitted Ivy students score 1530+.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is 1300 a good SAT score?&lt;/strong&gt;
Yes — 1300 is the 87th percentile. Competitive at the majority of four-year universities, including selective state schools.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Is 1100 a good SAT score?&lt;/strong&gt;
1100 is above the national average (60th percentile) but below the median for selective four-year colleges. Solid for less selective schools, but worth retaking if you’re aiming higher.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;What’s the average score on the SAT?&lt;/strong&gt;
About &lt;strong&gt;1050&lt;/strong&gt; (recent College Board data: ~525 R&amp;#x26;W + ~520 Math).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;What’s the lowest SAT score?&lt;/strong&gt;
The minimum possible score is &lt;strong&gt;400&lt;/strong&gt; (200 in each section). Almost no one actually scores that low — the bottom 1% of test takers score 750 or below.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Do colleges still require the SAT?&lt;/strong&gt;
A growing number of colleges are &lt;strong&gt;test-optional&lt;/strong&gt; (e.g., the entire UC system, plus most Ivies as of 2025), meaning you can choose not to submit scores. But submitting a strong score (above the school’s 50th percentile) still helps. Some schools — like MIT, Georgetown, Caltech — have returned to &lt;strong&gt;test-required&lt;/strong&gt; in 2024–2025.&lt;/p&gt;</content:encoded></item><item><title><![CDATA[SAT Test Dates 2026 (and Early 2027): Full Calendar + Registration Deadlines]]></title><description><![CDATA[Every Digital SAT test date for 2026 — international and US — plus registration deadlines, late fee cutoffs, and how to pick the right one for your timeline.]]></description><link>https://www.makon.ai/blog/sat-test-dates-2026</link><guid isPermaLink="false">https://www.makon.ai/blog/sat-test-dates-2026</guid><pubDate>Thu, 02 Apr 2026 00:00:00 GMT</pubDate><content:encoded>&lt;p&gt;If you’re planning your SAT prep, the first thing to nail down is the test date. Pick a date you can hit comfortably with a real study runway — not one that forces you to cram. This page lists every Digital SAT test date for 2026 (and the first few for 2027), the registration deadline for each, and a few rules of thumb for choosing.&lt;/p&gt;
&lt;p&gt;The Digital SAT is the only format College Board now offers worldwide, so all dates below are for the digital test. Score release happens about two weeks after test day.&lt;/p&gt;
&lt;h2 id=&quot;digital-sat-test-dates-2026--united-states&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#digital-sat-test-dates-2026--united-states&quot; aria-label=&quot;digital sat test dates 2026  united states permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Digital SAT test dates 2026 — United States&lt;/h2&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Test date&lt;/th&gt;
&lt;th&gt;Registration deadline&lt;/th&gt;
&lt;th&gt;Late registration&lt;/th&gt;
&lt;th&gt;Score release&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;March 14, 2026&lt;/td&gt;
&lt;td&gt;February 27&lt;/td&gt;
&lt;td&gt;March 4&lt;/td&gt;
&lt;td&gt;March 27&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;May 2, 2026&lt;/td&gt;
&lt;td&gt;April 17&lt;/td&gt;
&lt;td&gt;April 22&lt;/td&gt;
&lt;td&gt;May 15&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;June 6, 2026&lt;/td&gt;
&lt;td&gt;May 22&lt;/td&gt;
&lt;td&gt;May 27&lt;/td&gt;
&lt;td&gt;June 19&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;August 22, 2026&lt;/td&gt;
&lt;td&gt;August 7&lt;/td&gt;
&lt;td&gt;August 12&lt;/td&gt;
&lt;td&gt;September 4&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;September 12, 2026 &lt;em&gt;(school day)&lt;/em&gt;&lt;/td&gt;
&lt;td&gt;varies by school&lt;/td&gt;
&lt;td&gt;n/a&lt;/td&gt;
&lt;td&gt;September 25&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;October 3, 2026&lt;/td&gt;
&lt;td&gt;September 18&lt;/td&gt;
&lt;td&gt;September 23&lt;/td&gt;
&lt;td&gt;October 16&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;November 7, 2026&lt;/td&gt;
&lt;td&gt;October 23&lt;/td&gt;
&lt;td&gt;October 28&lt;/td&gt;
&lt;td&gt;November 20&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;December 5, 2026&lt;/td&gt;
&lt;td&gt;November 20&lt;/td&gt;
&lt;td&gt;November 25&lt;/td&gt;
&lt;td&gt;December 18&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Dates and deadlines are taken from College Board’s published Digital SAT calendar. Always confirm on &lt;a href=&quot;https://satsuite.collegeboard.org&quot;&gt;satsuite.collegeboard.org&lt;/a&gt; before booking — schools occasionally adjust school-day administrations.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;digital-sat-test-dates-2026--international&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#digital-sat-test-dates-2026--international&quot; aria-label=&quot;digital sat test dates 2026  international permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Digital SAT test dates 2026 — International&lt;/h2&gt;
&lt;p&gt;International dates mirror US weekend dates with a couple of regional substitutions:&lt;/p&gt;
&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Test date&lt;/th&gt;
&lt;th&gt;Registration deadline&lt;/th&gt;
&lt;th&gt;Notes&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;March 14, 2026&lt;/td&gt;
&lt;td&gt;February 27&lt;/td&gt;
&lt;td&gt;Most countries&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;May 2, 2026&lt;/td&gt;
&lt;td&gt;April 17&lt;/td&gt;
&lt;td&gt;Most countries&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;June 6, 2026&lt;/td&gt;
&lt;td&gt;May 22&lt;/td&gt;
&lt;td&gt;Most countries&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;August 22, 2026&lt;/td&gt;
&lt;td&gt;August 7&lt;/td&gt;
&lt;td&gt;Most countries&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;October 3, 2026&lt;/td&gt;
&lt;td&gt;September 18&lt;/td&gt;
&lt;td&gt;Most countries&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;November 7, 2026&lt;/td&gt;
&lt;td&gt;October 23&lt;/td&gt;
&lt;td&gt;Most countries&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;December 5, 2026&lt;/td&gt;
&lt;td&gt;November 20&lt;/td&gt;
&lt;td&gt;Most countries&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;h2 id=&quot;early-2027-sat-test-dates-preview&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#early-2027-sat-test-dates-preview&quot; aria-label=&quot;early 2027 sat test dates preview permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Early 2027 SAT test dates (preview)&lt;/h2&gt;
&lt;p&gt;If you’re a junior planning ahead, these are the next dates after the 2026 calendar:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;March 13, 2027&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;May 1, 2027&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;June 5, 2027&lt;/strong&gt;&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Registration usually opens 4–5 months before the test date.&lt;/p&gt;
&lt;h2 id=&quot;how-to-pick-the-right-sat-test-date&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-to-pick-the-right-sat-test-date&quot; aria-label=&quot;how to pick the right sat test date permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How to pick the right SAT test date&lt;/h2&gt;
&lt;p&gt;A few rules of thumb that beat “whatever date is next.”&lt;/p&gt;
&lt;h3 id=&quot;1-give-yourself-at-least-8-weeks-of-real-prep&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#1-give-yourself-at-least-8-weeks-of-real-prep&quot; aria-label=&quot;1 give yourself at least 8 weeks of real prep permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;1. Give yourself at least 8 weeks of real prep&lt;/h3&gt;
&lt;p&gt;If you’re starting from scratch, plan on 8–12 weeks of consistent study. Two weeks of cramming rarely moves the needle. Pick a test date that gives you that runway, not the soonest one available.&lt;/p&gt;
&lt;h3 id=&quot;2-take-the-sat-twice--first-in-spring-second-in-fall&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#2-take-the-sat-twice--first-in-spring-second-in-fall&quot; aria-label=&quot;2 take the sat twice  first in spring second in fall permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;2. Take the SAT twice — first in spring, second in fall&lt;/h3&gt;
&lt;p&gt;The cleanest pattern for college applications:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;First sitting:&lt;/strong&gt; spring of junior year (March, May, or June 2026 if you’re a 2027 grad). Low-stakes; you’ve still got time to retake.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Second sitting:&lt;/strong&gt; August or October of senior year. By now you know your weak spots and you’re fresh from summer prep.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;College admissions are usually best served by &lt;strong&gt;August or October scores&lt;/strong&gt; for early action / early decision deadlines (typically Nov 1).&lt;/p&gt;
&lt;h3 id=&quot;3-avoid-your-busiest-months&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#3-avoid-your-busiest-months&quot; aria-label=&quot;3 avoid your busiest months permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;3. Avoid your busiest months&lt;/h3&gt;
&lt;p&gt;If May is finals month, do not pick the May SAT. The exam doesn’t care that you’re tired. June (after AP exams) and August (before school) are the two underrated dates because they fall outside heavy school-load windows.&lt;/p&gt;
&lt;h3 id=&quot;4-school-day-sat-vs-weekend-sat&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#4-school-day-sat-vs-weekend-sat&quot; aria-label=&quot;4 school day sat vs weekend sat permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;4. School-day SAT vs. weekend SAT&lt;/h3&gt;
&lt;p&gt;Many US schools offer a free in-school administration in &lt;strong&gt;September, October, or April&lt;/strong&gt; (varies by district). It’s the same test and counts the same. The trade-off: you don’t pick the date, and you can’t retake it through your school until the next year. Use it as your &lt;strong&gt;first sitting&lt;/strong&gt; if your school offers it.&lt;/p&gt;
&lt;h3 id=&quot;5-match-the-date-to-your-application-deadlines-not-just-your-prep-timeline&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#5-match-the-date-to-your-application-deadlines-not-just-your-prep-timeline&quot; aria-label=&quot;5 match the date to your application deadlines not just your prep timeline permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;5. Match the date to your application deadlines, not just your prep timeline&lt;/h3&gt;
&lt;p&gt;Most students think only about how many weeks of study they have. The other half of the equation is when colleges actually need the score:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Early Action / Early Decision (typically Nov 1):&lt;/strong&gt; the latest test date that arrives in time is usually &lt;strong&gt;August or early October&lt;/strong&gt;. Score release is ~2 weeks after test day, then add another week for College Board to send to schools. November scores almost always miss EA/ED deadlines.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Regular Decision (typically Jan 1–Feb 1):&lt;/strong&gt; December and even January scores can still make it, depending on the school’s explicit cutoff. Always check each college’s testing policy page — some publish a hard &quot;scores must be received by&quot; date that is later than their app deadline.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Rolling admissions:&lt;/strong&gt; earlier scores are strictly better. Take it the first window you’re ready for.&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&quot;6-dont-schedule-back-to-back-sittings&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#6-dont-schedule-back-to-back-sittings&quot; aria-label=&quot;6 dont schedule back to back sittings permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;6. Don’t schedule back-to-back sittings&lt;/h3&gt;
&lt;p&gt;Booking the next test date 4 weeks after your first sitting feels productive — it’s usually a mistake. You won’t have your first score until ~2 weeks in, leaving only 2 weeks to actually study based on what you got wrong. The cleanest gap between sittings is &lt;strong&gt;8–12 weeks&lt;/strong&gt;: enough time to get the score, do a full diagnostic of weak areas, and run focused practice.&lt;/p&gt;
&lt;h2 id=&quot;bluebook-prep-timeline--what-to-do-before-test-day&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#bluebook-prep-timeline--what-to-do-before-test-day&quot; aria-label=&quot;bluebook prep timeline  what to do before test day permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Bluebook prep timeline — what to do before test day&lt;/h2&gt;
&lt;p&gt;The Digital SAT is delivered through Bluebook, College Board’s testing app. Treat the app like part of the test itself — fumbling the interface on test day costs real points.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;8+ weeks out:&lt;/strong&gt; download Bluebook on the device you’ll test on. It runs on Mac, Windows, iPad, and managed Chromebooks. Confirm your device meets the system requirements (minimum 3 hours of battery life, supported OS version).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;4 weeks out:&lt;/strong&gt; complete at least one full official practice test inside Bluebook. The interface — annotation tool, flag-for-review, the embedded Desmos calculator, the digital reference sheet — is identical to the real exam. The first time you use Desmos under timed conditions should not be on test day.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;1 week out:&lt;/strong&gt; open the &lt;strong&gt;Test Preview&lt;/strong&gt; in Bluebook for your assigned test date. This unlocks a few days before the exam and asks you to confirm device readiness. Skipping this is the most common reason students show up with broken setups.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Day-of:&lt;/strong&gt; charge to 100%, bring your charger anyway, screenshot your admission ticket as backup.&lt;/p&gt;
&lt;p&gt;If you’re testing on a school-issued device, check with your school’s IT desk: some districts pre-install Bluebook, others require the student to install it themselves. The student is the one who gets locked out at 8am if it isn’t done.&lt;/p&gt;
&lt;h2 id=&quot;registration-how-to-actually-book-the-test&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#registration-how-to-actually-book-the-test&quot; aria-label=&quot;registration how to actually book the test permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Registration: how to actually book the test&lt;/h2&gt;
&lt;ol&gt;
&lt;li&gt;Create a College Board account at &lt;a href=&quot;https://satsuite.collegeboard.org&quot;&gt;satsuite.collegeboard.org&lt;/a&gt; if you don’t have one.&lt;/li&gt;
&lt;li&gt;Choose a test date and test center near you. Center capacity fills early in big cities — book 6+ weeks out.&lt;/li&gt;
&lt;li&gt;Pay the fee (US: $68 for the standard SAT; international: ~$117). Fee waivers are available for eligible students.&lt;/li&gt;
&lt;li&gt;Bring your admission ticket + a photo ID + a charged personal device (laptop or tablet) on test day.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;Late registration adds ~$34. Standby testing is no longer offered for the Digital SAT.&lt;/p&gt;
&lt;h2 id=&quot;how-to-actually-get-a-seat--test-center-capacity&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#how-to-actually-get-a-seat--test-center-capacity&quot; aria-label=&quot;how to actually get a seat  test center capacity permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;How to actually get a seat — test center capacity&lt;/h2&gt;
&lt;p&gt;Test centers in big metros (NYC, LA, Bay Area, Boston, DC, Atlanta) routinely fill up 6–8 weeks before the exam. By the time the registration deadline arrives, the only seats left are 30–60 minutes from your home. Strategy:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Register the week registration opens&lt;/strong&gt;, not the week the deadline closes. Pick your two closest centers as primary + backup.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;If your closest centers are full&lt;/strong&gt;, widen your radius to ~25 miles before changing test dates. A longer drive on test morning is almost always better than pushing the date by 2 months.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Suburbs of big metros fill up last&lt;/strong&gt;. If downtown and inner suburbs are gone, look at outer suburbs and exurbs — the same district often has high schools 30 miles out with seats left.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Some test centers reopen seats&lt;/strong&gt; as confirmed test takers cancel or get reassigned. Check the College Board portal weekly until ~2 weeks out.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;The school-day SAT is your safety net.&lt;/strong&gt; If your school offers in-school SAT, that seat is guaranteed without using the regional center pool.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;If your school doesn’t offer a school-day SAT and your area is genuinely full, College Board occasionally opens &lt;strong&gt;additional test dates&lt;/strong&gt; in high-demand metros. They’re announced 4–6 weeks ahead — sign up for College Board email alerts so you don’t miss them.&lt;/p&gt;
&lt;h2 id=&quot;testing-with-accommodations--different-timeline&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#testing-with-accommodations--different-timeline&quot; aria-label=&quot;testing with accommodations  different timeline permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Testing with accommodations — different timeline&lt;/h2&gt;
&lt;p&gt;If you qualify for accommodations (extended time, breaks, screen reader, etc.), the timeline is &lt;strong&gt;completely different&lt;/strong&gt; from standard registration. You don’t register first and then request accommodations — the accommodation has to be approved before you can register for a specific date.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Apply through Services for Students with Disabilities (SSD)&lt;/strong&gt; at least &lt;strong&gt;7 weeks before&lt;/strong&gt; your target test date. Your school’s SSD coordinator submits the request; you provide supporting documentation.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Approval typically takes 7 weeks.&lt;/strong&gt; If you’re approved for College Board accommodations, the same approval covers AP exams and PSAT, so it’s reusable across multiple tests.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Once approved, you register normally&lt;/strong&gt; at satsuite.collegeboard.org — your accommodations attach to your account automatically.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;If denied or partially approved&lt;/strong&gt;, you can appeal, but the appeal adds another 4–6 weeks. Plan for the test date &lt;em&gt;after&lt;/em&gt; the one you originally targeted.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;If you already have an IEP or 504 plan at school, your SSD coordinator should be able to upload your existing documentation. You generally don’t need fresh evaluations — just current documentation that matches the accommodation you’re requesting.&lt;/p&gt;
&lt;h2 id=&quot;international-students--whats-different&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#international-students--whats-different&quot; aria-label=&quot;international students  whats different permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;International students — what’s different&lt;/h2&gt;
&lt;p&gt;If you’re testing outside the US, three things change:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Fewer dates.&lt;/strong&gt; The August and September school-day administrations are US-only. International dates are March, May, June, October, November, and December.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Higher fee.&lt;/strong&gt; International registration is ~$117 (vs $68 in the US) plus a regional surcharge in some countries (India, mainland China, South Korea).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Earlier registration close.&lt;/strong&gt; Centers in India and China routinely fill 8–10 weeks before the test, not 4. Register the day registration opens for your test date.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;The score release timeline is identical to the US (~2 weeks after test day). College Board treats international scores identically to domestic scores for admissions — there’s no separate &quot;international SAT.&quot;&lt;/p&gt;
&lt;h2 id=&quot;what-to-do-once-you-have-a-date&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#what-to-do-once-you-have-a-date&quot; aria-label=&quot;what to do once you have a date permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;What to do once you have a date&lt;/h2&gt;
&lt;p&gt;You picked a date. Now:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Take a full-length practice test this week.&lt;/strong&gt; It’s your baseline. Use our &lt;a href=&quot;/sat-score-calculator&quot;&gt;free Digital SAT score calculator&lt;/a&gt; to convert your raw correct answers into a scaled score so you know where you’re starting.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Map your study plan backward from test day.&lt;/strong&gt; Eight weeks out, focus on weak content. Four weeks out, switch to timed practice. Final week: light review only.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Practice on the actual platform.&lt;/strong&gt; The Digital SAT is delivered through College Board’s Bluebook app — install it now and run the official practice tests so the interface isn’t a surprise on test day.&lt;/li&gt;
&lt;/ol&gt;
&lt;h2 id=&quot;5-common-mistakes-when-picking-a-test-date&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#5-common-mistakes-when-picking-a-test-date&quot; aria-label=&quot;5 common mistakes when picking a test date permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;5 common mistakes when picking a test date&lt;/h2&gt;
&lt;p&gt;After thousands of students, the same mistakes repeat:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Picking the soonest date because &quot;I want to get it done.&quot;&lt;/strong&gt; Two weeks of cramming almost never moves your score 100+ points. Pick the date you can actually be ready for, not the closest one.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Testing during AP / IB / finals month.&lt;/strong&gt; May is the worst SAT date for most US students because AP exams happen the same week. The brain has a fixed reservoir of focus per month — don’t spend it twice.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Booking only one sitting.&lt;/strong&gt; Most score growth happens between sittings, when you finally know what you got wrong. Plan two from the start; treat the first as a real practice run.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Waiting for the score before booking the next date.&lt;/strong&gt; By the time the score arrives, the registration deadline for the next date may have already passed. Book both dates up front; cancellation refunds are partial but available.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Confusing the registration deadline with the late deadline.&lt;/strong&gt; The &quot;late&quot; deadline costs ~$34 extra and offers fewer center options. The standard deadline is the one to plan around.&lt;/li&gt;
&lt;/ol&gt;
&lt;h2 id=&quot;day-before--day-of-checklist&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#day-before--day-of-checklist&quot; aria-label=&quot;day before  day of checklist permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Day-before / day-of checklist&lt;/h2&gt;
&lt;p&gt;The night before:&lt;/p&gt;
&lt;ul class=&quot;contains-task-list&quot;&gt;
&lt;li class=&quot;task-list-item&quot;&gt;&lt;input type=&quot;checkbox&quot; disabled&gt; Bluebook installed, latest version, fully updated&lt;/li&gt;
&lt;li class=&quot;task-list-item&quot;&gt;&lt;input type=&quot;checkbox&quot; disabled&gt; Device charged to 100%, charger packed&lt;/li&gt;
&lt;li class=&quot;task-list-item&quot;&gt;&lt;input type=&quot;checkbox&quot; disabled&gt; Admission ticket printed (and screenshotted as backup)&lt;/li&gt;
&lt;li class=&quot;task-list-item&quot;&gt;&lt;input type=&quot;checkbox&quot; disabled&gt; Photo ID located (driver’s license, passport, or school ID with photo)&lt;/li&gt;
&lt;li class=&quot;task-list-item&quot;&gt;&lt;input type=&quot;checkbox&quot; disabled&gt; Approved calculator if you’re bringing one (Desmos is built into Bluebook, but bring a physical TI calculator as backup)&lt;/li&gt;
&lt;li class=&quot;task-list-item&quot;&gt;&lt;input type=&quot;checkbox&quot; disabled&gt; Snack + water bottle for the break&lt;/li&gt;
&lt;li class=&quot;task-list-item&quot;&gt;&lt;input type=&quot;checkbox&quot; disabled&gt; Plan your route — give yourself a 30-minute buffer&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;The morning of: eat protein, leave 30 minutes earlier than you think you need to, arrive by check-in time (not start time). Test centers turn away latecomers and don’t reschedule for free.&lt;/p&gt;
&lt;h2 id=&quot;faqs&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#faqs&quot; aria-label=&quot;faqs permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;FAQs&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;Can I take the SAT more than once?&lt;/strong&gt;
Yes — most students take it twice, some three times. Colleges typically superscore (take your highest section scores across sittings), so retaking is usually a positive expected value if you’ve studied.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;When should I take my first SAT?&lt;/strong&gt;
Spring of junior year is the standard answer (March, May, or June). That gives you summer prep + a fall retake before applications.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;What if I miss the registration deadline?&lt;/strong&gt;
You have a &lt;del&gt;5-day late registration window for an extra fee (&lt;/del&gt;$34). After that you have to wait for the next date.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Are there fee waivers?&lt;/strong&gt;
Yes — eligible juniors and seniors can get up to &lt;strong&gt;two free SATs&lt;/strong&gt; plus free score reports. Ask your school counselor.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;How long does the Digital SAT take?&lt;/strong&gt;
2 hours and 14 minutes of testing time + a 10-minute break. Plus check-in (~30 minutes). Plan for ~3 hours total at the test center.&lt;/p&gt;</content:encoded></item><item><title><![CDATA[SAT Math Formula Sheet (Digital SAT 2026): Every Formula You Need]]></title><description><![CDATA[The complete Digital SAT math formula sheet. The 13 reference formulas you’re given on test day, plus the 30+ formulas you need to memorize.]]></description><link>https://www.makon.ai/blog/sat-math-formulas</link><guid isPermaLink="false">https://www.makon.ai/blog/sat-math-formulas</guid><pubDate>Sun, 15 Mar 2026 00:00:00 GMT</pubDate><content:encoded>&lt;p&gt;The Digital SAT gives you a built-in reference sheet on test day — but only &lt;strong&gt;13 formulas&lt;/strong&gt;. Everything else, you’re expected to know cold. This guide is your complete SAT math cheat sheet: the formulas the College Board gives you, the ones they don’t (but you absolutely need), and a quick example for each.&lt;/p&gt;
&lt;p&gt;Bookmark this page. By test day, every formula here should be muscle memory.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Tip:&lt;/strong&gt; practice using these formulas, don’t just memorize them. Plug them into a real Digital SAT practice question on Makon AI’s &lt;a href=&quot;https://app.makon.ai&quot;&gt;free Question Bank&lt;/a&gt; and see your scaled score on our &lt;a href=&quot;/sat-score-calculator&quot;&gt;Digital SAT score calculator&lt;/a&gt;.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;part-1--the-13-formulas-the-sat-gives-you-the-official-reference-sheet&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#part-1--the-13-formulas-the-sat-gives-you-the-official-reference-sheet&quot; aria-label=&quot;part 1  the 13 formulas the sat gives you the official reference sheet permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Part 1 — The 13 formulas the SAT gives you (the official reference sheet)&lt;/h2&gt;
&lt;p&gt;These show up at the start of each Math module. You don’t need to memorize them, but you do need to know &lt;strong&gt;when to use each one&lt;/strong&gt;.&lt;/p&gt;
&lt;h3 id=&quot;geometry--area--circumference&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#geometry--area--circumference&quot; aria-label=&quot;geometry  area  circumference permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Geometry — area &amp;#x26; circumference&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;1. Area of a rectangle&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;A = lw&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02691em;&quot;&gt;lw&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;2. Area of a triangle&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;A = \tfrac{1}{2}bh&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1901em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8451em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;bh&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;3. Pythagorean theorem&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;a^2 + b^2 = c^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8974em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;4. Special right triangle (45-45-90)&lt;/strong&gt; — sides in ratio &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msqrt&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;1 : 1 : \sqrt{2}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.04em;vertical-align:-0.1328em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord sqrt&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9072em;&quot;&gt;&lt;span class=&quot;svg-align&quot; style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot; style=&quot;padding-left:0.833em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-2.8672em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;hide-tail&quot; style=&quot;min-width:0.853em;height:1.08em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;400em&quot; height=&quot;1.08em&quot; viewBox=&quot;0 0 400000 1080&quot; preserveAspectRatio=&quot;xMinYMin slice&quot;&gt;&lt;path d=&quot;M95,702
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&lt;p&gt;&lt;strong&gt;5. Special right triangle (30-60-90)&lt;/strong&gt; — sides in ratio &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msqrt&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msqrt&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;1 : \sqrt{3} : 2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.04em;vertical-align:-0.1328em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord sqrt&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9072em;&quot;&gt;&lt;span class=&quot;svg-align&quot; style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot; style=&quot;padding-left:0.833em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-2.8672em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;hide-tail&quot; style=&quot;min-width:0.853em;height:1.08em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;400em&quot; height=&quot;1.08em&quot; viewBox=&quot;0 0 400000 1080&quot; preserveAspectRatio=&quot;xMinYMin slice&quot;&gt;&lt;path d=&quot;M95,702
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&lt;p&gt;&lt;strong&gt;6. Circle: area&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;A = \pi r^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; · &lt;strong&gt;circumference&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;C = 2\pi r&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.07153em;&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;7. Cylinder volume&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;V = \pi r^2 h&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.22222em;&quot;&gt;V&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;8. Sphere volume&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;V = \tfrac{4}{3}\pi r^3&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.22222em;&quot;&gt;V&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1901em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8451em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;9. Cone volume&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;V = \tfrac{1}{3}\pi r^2 h&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.22222em;&quot;&gt;V&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1901em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8451em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;10. Pyramid volume&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;V = \tfrac{1}{3}lwh&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.22222em;&quot;&gt;V&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1901em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8451em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02691em;&quot;&gt;lw&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;11. Rectangular prism volume&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;V = lwh&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.22222em;&quot;&gt;V&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02691em;&quot;&gt;lw&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;12. Number of degrees in a circle&lt;/strong&gt; = 360&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;13. Sum of angles in a triangle&lt;/strong&gt; = 180°&lt;/p&gt;
&lt;p&gt;That’s the entire reference sheet. If you’re relying on it for anything more complex, you’re wasting time.&lt;/p&gt;
&lt;h2 id=&quot;part-2--formulas-you-need-to-memorize-the-sat-does-not-give-you-these&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#part-2--formulas-you-need-to-memorize-the-sat-does-not-give-you-these&quot; aria-label=&quot;part 2  formulas you need to memorize the sat does not give you these permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Part 2 — Formulas you need to memorize (the SAT does NOT give you these)&lt;/h2&gt;
&lt;p&gt;This is where most of the points are won or lost. The SAT loves to test these because they’re “high school knowledge,” not lookup-able.&lt;/p&gt;
&lt;h3 id=&quot;algebra--linear-equations&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#algebra--linear-equations&quot; aria-label=&quot;algebra  linear equations permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Algebra &amp;#x26; linear equations&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Slope of a line&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;m = \frac{y_2 - y_1}{x_2 - x_1}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2995em;vertical-align:-0.4451em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8544em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3173em;&quot;&gt;&lt;span style=&quot;top:-2.357em;margin-left:0em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.143em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3173em;&quot;&gt;&lt;span style=&quot;top:-2.357em;margin-left:0em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.143em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.4461em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3173em;&quot;&gt;&lt;span style=&quot;top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.143em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3173em;&quot;&gt;&lt;span style=&quot;top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.143em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.4451em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Slope-intercept form&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;y = mx + b&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Point-slope form&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;y - y_1 = m(x - x_1)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7778em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Standard form&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;ax + by = c&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;c&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Distance between two points&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.24em;vertical-align:-0.305em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord sqrt&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.935em;&quot;&gt;&lt;span class=&quot;svg-align&quot; style=&quot;top:-3.2em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.2em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot; style=&quot;padding-left:1em;&quot;&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.7401em;&quot;&gt;&lt;span style=&quot;top:-2.989em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.7401em;&quot;&gt;&lt;span style=&quot;top:-2.989em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-2.895em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.2em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;hide-tail&quot; style=&quot;min-width:1.02em;height:1.28em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; 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&lt;p&gt;&lt;strong&gt;Midpoint of a segment&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo fence=&quot;true&quot;&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo fence=&quot;true&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.10903em;&quot;&gt;M&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2044em;vertical-align:-0.35em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;minner&quot;&gt;&lt;span class=&quot;mopen delimcenter&quot; style=&quot;top:0em;&quot;&gt;&lt;span class=&quot;delimsizing size1&quot;&gt;(&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8184em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.4101em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3173em;&quot;&gt;&lt;span style=&quot;top:-2.357em;margin-left:0em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.143em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3173em;&quot;&gt;&lt;span style=&quot;top:-2.357em;margin-left:0em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.143em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8544em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.4461em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3173em;&quot;&gt;&lt;span style=&quot;top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.143em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3173em;&quot;&gt;&lt;span style=&quot;top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.143em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose delimcenter&quot; style=&quot;top:0em;&quot;&gt;&lt;span class=&quot;delimsizing size1&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Parallel lines&lt;/strong&gt; have equal slopes. &lt;strong&gt;Perpendicular lines&lt;/strong&gt; have slopes that are negative reciprocals: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;m_1 \cdot m_2 = -1&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5945em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;.&lt;/p&gt;
&lt;h3 id=&quot;quadratics&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#quadratics&quot; aria-label=&quot;quadratics permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Quadratics&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Standard form&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;y = ax^2 + bx + c&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8974em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7778em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;c&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Vertex form&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;y = a(x - h)^2 + k&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0641em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03148em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — vertex at &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;(h, k)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03148em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Factored form&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;y = a(x - r_1)(x - r_2)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — roots at &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;r_1, r_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Quadratic formula&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.3845em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.0395em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;±&lt;/span&gt;&lt;span class=&quot;mord sqrt mtight&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9221em;&quot;&gt;&lt;span class=&quot;svg-align&quot; style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mtight&quot; style=&quot;padding-left:0.833em;&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.7463em;&quot;&gt;&lt;span style=&quot;top:-2.786em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;c&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-2.8821em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;hide-tail mtight&quot; style=&quot;min-width:0.853em;height:1.08em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;400em&quot; height=&quot;1.08em&quot; viewBox=&quot;0 0 400000 1080&quot; preserveAspectRatio=&quot;xMinYMin slice&quot;&gt;&lt;path d=&quot;M95,702
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M834 80h400000v40h-400000z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1179em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Discriminant&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;b^2 - 4ac&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8974em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;c&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&gt; 0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5782em;vertical-align:-0.0391em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;: two real roots&lt;/li&gt;
&lt;li&gt;&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;= 0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.3669em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;: one real root&lt;/li&gt;
&lt;li&gt;&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x3C;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&amp;#x3C; 0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5782em;vertical-align:-0.0391em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;&amp;#x3C;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;: no real roots&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Vertex from standard form&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x = -\frac{b}{2a}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2251em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8801em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Sum of roots&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;= -\frac{b}{a}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.3669em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2251em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8801em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; · &lt;strong&gt;Product of roots&lt;/strong&gt; &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;= \frac{c}{a}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.3669em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0404em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6954em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;c&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;h3 id=&quot;exponents--radicals&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#exponents--radicals&quot; aria-label=&quot;exponents  radicals permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Exponents &amp;#x26; radicals&lt;/h3&gt;
&lt;p&gt;&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x^a \cdot x^b = x^{a+b}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6644em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6644em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\frac{x^a}{x^b} = x^{a-b}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2684em;vertical-align:-0.3574em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.911em;&quot;&gt;&lt;span style=&quot;top:-2.6426em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.782em;&quot;&gt;&lt;span style=&quot;top:-2.786em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.7385em;&quot;&gt;&lt;span style=&quot;top:-2.931em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3574em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;(x^a)^b = x^{ab}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0991em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6644em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8491em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;ab&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x^{-a} = \frac{1}{x^a}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7713em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.7713em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1901em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8451em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.5935em;&quot;&gt;&lt;span style=&quot;top:-2.786em;margin-right:0.0714em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size3 size1 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mroot&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mroot&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x^{1/n} = \sqrt[n]{x}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.888em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.888em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1/&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.04em;vertical-align:-0.2397em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord sqrt&quot;&gt;&lt;span class=&quot;root&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.5516em;&quot;&gt;&lt;span style=&quot;top:-2.8363em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size1 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8003em;&quot;&gt;&lt;span class=&quot;svg-align&quot; style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot; style=&quot;padding-left:0.833em;&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-2.7603em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;hide-tail&quot; style=&quot;min-width:0.853em;height:1.08em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;400em&quot; height=&quot;1.08em&quot; viewBox=&quot;0 0 400000 1080&quot; preserveAspectRatio=&quot;xMinYMin slice&quot;&gt;&lt;path d=&quot;M95,702
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&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x^0 = 1&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; (when &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo mathvariant=&quot;normal&quot;&gt;≠&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x \neq 0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;&lt;span class=&quot;mrel&quot;&gt;&lt;span class=&quot;mord vbox&quot;&gt;&lt;span class=&quot;thinbox&quot;&gt;&lt;span class=&quot;rlap&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;inner&quot;&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mrel&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;fix&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;)&lt;/p&gt;
&lt;h3 id=&quot;percentages-ratios--proportions&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#percentages-ratios--proportions&quot; aria-label=&quot;percentages ratios  proportions permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Percentages, ratios &amp;#x26; proportions&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Percent change&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mtext&gt;percent change&lt;/mtext&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mtext&gt;new&lt;/mtext&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mtext&gt;old&lt;/mtext&gt;&lt;/mrow&gt;&lt;mtext&gt;old&lt;/mtext&gt;&lt;/mfrac&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord text&quot;&gt;&lt;span class=&quot;mord&quot;&gt;percent change&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2251em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8801em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;old&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;new&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;old&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;100&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Percent of a number&lt;/strong&gt;: convert percent to decimal, multiply.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;25% of 80 = 0.25 × 80 = 20&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Direct proportion&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;y = kx&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03148em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — y goes up when x goes up.
&lt;strong&gt;Inverse proportion&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;y = \frac{k}{x}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2251em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8801em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.03148em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — y goes down when x goes up.&lt;/p&gt;
&lt;h3 id=&quot;functions&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#functions&quot; aria-label=&quot;functions permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Functions&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Function notation&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f(x)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.10764em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; means “the value of f at x”&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f(g(x))&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.10764em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;))&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — plug &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;g(x)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;g&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; in everywhere &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.10764em;&quot;&gt;f&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; has &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Linear function&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f(x) = mx + b&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.10764em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;strong&gt;Exponential growth&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f(x) = a(1 + r)^x&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.10764em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6644em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;strong&gt;Exponential decay&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f(x) = a(1 - r)^x&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.10764em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6644em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;h3 id=&quot;statistics--data&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#statistics--data&quot; aria-label=&quot;statistics  data permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Statistics &amp;#x26; data&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Mean (average)&lt;/strong&gt; = sum / count&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Median&lt;/strong&gt; = middle value when sorted&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Mode&lt;/strong&gt; = most frequent value&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Range&lt;/strong&gt; = max − min&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Standard deviation&lt;/strong&gt; — you don’t calculate it on the SAT, but you should know that &lt;strong&gt;higher SD = data more spread out&lt;/strong&gt;.&lt;/p&gt;
&lt;h3 id=&quot;probability&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#probability&quot; aria-label=&quot;probability permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Probability&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Single event probability&lt;/strong&gt;
&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mtext&gt;favorable outcomes&lt;/mtext&gt;&lt;mtext&gt;total outcomes&lt;/mtext&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;P = \frac{\text{favorable outcomes}}{\text{total outcomes}}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.13889em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2251em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8801em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;total outcomes&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;favorable outcomes&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Independent events&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mtext&gt; and &lt;/mtext&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;P(A \text{ and } B) = P(A) \cdot P(B)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.13889em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mord text&quot;&gt;&lt;span class=&quot;mord&quot;&gt; and &lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.05017em;&quot;&gt;B&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.13889em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.13889em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.05017em;&quot;&gt;B&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Mutually exclusive events&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mtext&gt; or &lt;/mtext&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;P(A \text{ or } B) = P(A) + P(B)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.13889em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mord text&quot;&gt;&lt;span class=&quot;mord&quot;&gt; or &lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.05017em;&quot;&gt;B&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.13889em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.13889em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.05017em;&quot;&gt;B&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;h3 id=&quot;geometry--trigonometry-extras-the-reference-sheet-skips&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#geometry--trigonometry-extras-the-reference-sheet-skips&quot; aria-label=&quot;geometry  trigonometry extras the reference sheet skips permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Geometry &amp;#x26; trigonometry (extras the reference sheet skips)&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Area of a trapezoid&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;A = \tfrac{1}{2}(b_1 + b_2)h&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1901em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8451em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;b&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Area of a parallelogram&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;A = bh&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;bh&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Circle equation&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;(x - h)^2 + (y - k)^2 = r^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0641em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0641em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03148em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — center &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;(h,k)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03148em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, radius &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;r&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Arc length&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mtext&gt;arc&lt;/mtext&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mn&gt;360&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\text{arc} = \frac{\theta}{360} \cdot 2\pi r&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord text&quot;&gt;&lt;span class=&quot;mord&quot;&gt;arc&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2251em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8801em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;360&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Sector area&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mn&gt;360&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;A = \frac{\theta}{360} \cdot \pi r^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2251em;vertical-align:-0.345em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8801em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;360&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.03588em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Trig (right triangles)&lt;/strong&gt;:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mtext&gt;opposite&lt;/mtext&gt;&lt;mtext&gt;hypotenuse&lt;/mtext&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.3947em;vertical-align:-0.4811em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9136em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;hypotenuse&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.4461em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;opposite&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.4811em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/li&gt;
&lt;li&gt;&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mtext&gt;adjacent&lt;/mtext&gt;&lt;mtext&gt;hypotenuse&lt;/mtext&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.4133em;vertical-align:-0.4811em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9322em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;hypotenuse&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.4461em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;adjacent&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.4811em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/li&gt;
&lt;li&gt;&lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;tan&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mtext&gt;opposite&lt;/mtext&gt;&lt;mtext&gt;adjacent&lt;/mtext&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;tan&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.3947em;vertical-align:-0.4811em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9136em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;adjacent&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.4461em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord text mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;opposite&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.4811em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; (mnemonic: SOH-CAH-TOA)&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Trig identity&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\sin^2(\theta) + \cos^2(\theta) = 1&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1219em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8719em;&quot;&gt;&lt;span style=&quot;top:-3.1208em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0641em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Complementary angles&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;90&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;°&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\sin(\theta) = \cos(90° - \theta)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;90°&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;h3 id=&quot;sequences&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#sequences&quot; aria-label=&quot;sequences permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Sequences&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Arithmetic sequence&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;a_n = a_1 + (n - 1)d&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;d&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Geometric sequence&lt;/strong&gt;: &lt;span class=&quot;math math-inline&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;a_n = a_1 \cdot r^{n-1}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5945em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.02778em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;h2 id=&quot;part-3--how-to-actually-memorize-these&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#part-3--how-to-actually-memorize-these&quot; aria-label=&quot;part 3  how to actually memorize these permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Part 3 — How to actually memorize these&lt;/h2&gt;
&lt;p&gt;Don’t print the sheet and stare at it. Memory works through &lt;strong&gt;retrieval&lt;/strong&gt;, not rereading. Use this sequence:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Day 1:&lt;/strong&gt; read each formula once, write a 1-line example for each.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Day 2:&lt;/strong&gt; flashcards (front: name. back: formula). 10 minutes.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Day 3+:&lt;/strong&gt; every time you do a math practice question, &lt;strong&gt;identify which formula applies before solving&lt;/strong&gt;. That’s the skill the SAT tests.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;Inside Makon AI, the AI tutor flags which formula a question relies on whenever you miss it — that single feedback loop is worth more than re-reading this sheet.&lt;/p&gt;
&lt;h2 id=&quot;quick-reference-formulas-by-digital-sat-topic-area&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#quick-reference-formulas-by-digital-sat-topic-area&quot; aria-label=&quot;quick reference formulas by digital sat topic area permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Quick reference: formulas by Digital SAT topic area&lt;/h2&gt;
&lt;p&gt;The Digital SAT Math section covers four content areas. Here’s which formulas cluster where:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Algebra (~35% of questions):&lt;/strong&gt; linear equations, slope, systems, point-slope/standard/slope-intercept forms&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Advanced Math (~35%):&lt;/strong&gt; quadratics, exponents, function notation, polynomials&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Problem-Solving &amp;#x26; Data Analysis (~15%):&lt;/strong&gt; percent change, ratios, proportion, mean/median/mode, probability&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Geometry &amp;#x26; Trigonometry (~15%):&lt;/strong&gt; the reference-sheet formulas + circle equation, arc/sector, trig&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&quot;download-the-formula-sheet-pdf&quot; style=&quot;position:relative;&quot;&gt;&lt;a href=&quot;#download-the-formula-sheet-pdf&quot; aria-label=&quot;download the formula sheet pdf permalink&quot; class=&quot;anchor before&quot;&gt;&lt;svg aria-hidden=&quot;true&quot; focusable=&quot;false&quot; height=&quot;16&quot; version=&quot;1.1&quot; viewBox=&quot;0 0 16 16&quot; width=&quot;16&quot;&gt;&lt;path fill-rule=&quot;evenodd&quot; d=&quot;M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/a&gt;Download the formula sheet (PDF)&lt;/h2&gt;
&lt;p&gt;Coming soon — we’re putting the cheat-sheet version of this page on a one-page printable PDF. Bookmark this article so you can grab it when it drops.&lt;/p&gt;</content:encoded></item></channel></rss>