SOLID STATE PRESS
← Back to catalog
Quadratic Equations for the SAT and ACT cover
Coming soon
Coming soon to Amazon
This title is in our publishing queue.
Browse available titles
Mathematics

Quadratic Equations for the SAT and ACT

A High School Primer for Test Day

The quadratic unit is one of the most reliably tested topics on the SAT and ACT — and also one of the most likely to trip students up under timed conditions. Whether you blanked on the quadratic formula, never quite understood what the discriminant tells you, or need a fast refresher before test day, this guide cuts straight to what matters.

**TLDR: Quadratic Equations for the SAT and ACT** covers every form of the quadratic — standard, factored, and vertex — along with the solving techniques that show up most on real exams: factoring, the square root method, the quadratic formula, and completing the square. You'll learn how to read the discriminant to count solutions in seconds, how to find a parabola's vertex and axis of symmetry without graphing, and how to translate classic word problems into equations you can actually solve. A final section connects it all to graphs and quadratic-linear systems, the style of multi-step problem the SAT loves to end with.

This is a focused quadratic formula practice guide for SAT math and ACT math — not a 500-page textbook. It's written for high school students in grades 9–12 and early college students who need to get oriented fast, work through concrete examples, and walk into an exam with confidence. Parents helping their kids and tutors prepping a session will find it equally useful.

Pick it up, work through it in an afternoon, and know your quadratics cold.

What you'll learn
  • Recognize a quadratic equation in standard, factored, and vertex form, and convert between them
  • Solve quadratics quickly using factoring, the square root method, completing the square, and the quadratic formula
  • Use the discriminant to determine the number and type of solutions without solving
  • Read graphs of parabolas to find roots, vertex, axis of symmetry, and minimum or maximum values
  • Translate SAT/ACT word problems into quadratic equations and pick the right solution method under time pressure
What's inside
  1. 1. What Is a Quadratic Equation?
    Defines quadratics, introduces standard, factored, and vertex forms, and shows what the graph looks like.
  2. 2. Solving by Factoring and the Square Root Method
    Covers the two fastest solving techniques and when each one is the right tool on a timed test.
  3. 3. The Quadratic Formula and the Discriminant
    Teaches the universal solving tool and how to use the discriminant to count solutions instantly.
  4. 4. Completing the Square and Vertex Form
    Shows how to rewrite quadratics to find the vertex, axis of symmetry, and max/min values the SAT loves to test.
  5. 5. Graphs, Systems, and Word Problems on the SAT/ACT
    Connects equations to graphs, solves quadratic-linear systems, and translates classic word problems into equations.
Published by Solid State Press
Quadratic Equations for the SAT and ACT cover
TLDR STUDY GUIDES

Quadratic Equations for the SAT and ACT

A High School Primer for Test Day
Solid State Press

Quadratic Equations for the SAT and ACT

A High School Primer for Test Day

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Who This Book Is For

If you're staring down the math section of the SAT or ACT, taking Algebra 2 and hitting a wall on parabolas, or pulling together a last-minute review before a unit exam, this book was written for you. It works equally well as a standalone primer and as a companion to a full SAT math algebra study guide or a larger test-prep course.

This is a focused ACT math quadratics quick review book and an equally sharp SAT prep tool — covering factoring quadratics for a high school review, the quadratic formula with SAT-style practice, completing the square, vertex form, discriminants, and word problems drawn from real exam patterns. Think of it as a high school Algebra 2 quadratics primer stripped to what actually appears on test day. About 15 pages, no padding.

Read straight through once, pausing to work every example yourself before reading the solution. Then hit the practice problems at the end — that's where the quadratic equations SAT and ACT test prep skills get locked in.

Contents

  1. 1 What Is a Quadratic Equation?
  2. 2 Solving by Factoring and the Square Root Method
  3. 3 The Quadratic Formula and the Discriminant
  4. 4 Completing the Square and Vertex Form
  5. 5 Graphs, Systems, and Word Problems on the SAT/ACT
Chapter 1

What Is a Quadratic Equation?

Every quadratic equation is built from the same blueprint: a variable squared, usually some multiple of that variable, and a constant, all set equal to zero (or to another expression). More precisely, a quadratic equation is any equation that can be written in the form

$ax^2 + bx + c = 0$

where $a$, $b$, and $c$ are real numbers and $a \neq 0$. That last condition matters — if $a$ were zero, the $x^2$ term would vanish and you'd have a linear equation, not a quadratic.

The numbers $a$, $b$, and $c$ each have a name. The number $a$ is the leading coefficient (it sits in front of $x^2$), $b$ is the coefficient of $x$, and $c$ is the constant term. When you see something like $3x^2 - 5x + 2 = 0$, you can read off $a = 3$, $b = -5$, and $c = 2$ immediately. Getting comfortable with that identification is the first micro-skill to own.

The Three Forms You Will See

Quadratic equations appear on the SAT and ACT in three distinct disguises. Recognizing which form you're looking at tells you immediately which technique to reach for.

Standard form is what you just saw: $ax^2 + bx + c = 0$. It's the default, the form every other form can be expanded into, and the starting point for the quadratic formula and factoring.

Factored form writes the quadratic as a product of two linear factors:

$a(x - r_1)(x - r_2) = 0$

The values $r_1$ and $r_2$ are called the roots (also called solutions or zeros) — they are the $x$-values that make the equation equal zero. Factored form hands you the roots directly without any further algebra. If you see $(x - 3)(x + 5) = 0$, the roots are $x = 3$ and $x = -5$ by inspection.

Vertex form writes the quadratic as

$a(x - h)^2 + k = 0$

or, when describing the related function, $f(x) = a(x - h)^2 + k$. The point $(h, k)$ is the vertex — the tip of the U-shaped curve. Vertex form is the tool the SAT uses when it asks about maximum or minimum values, because $k$ is that maximum or minimum directly. Section 4 walks through exactly how to convert into this form.

Keep reading

You've read the first half of Chapter 1. The complete book covers 5 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon