Learn on PengiCalifornia Reveal Math, Algebra 1Unit 10: Quadratic Functions

10-4 Solving Quadratic Equations by Factoring

In this Grade 9 California Reveal Math Algebra 1 lesson, students learn to solve quadratic equations using the Square Root Property, the Zero Product Property, and factoring techniques including GCF factoring, trinomial factoring, and difference of squares. Students practice applying these methods to equations in standard form and connect the roots of an equation to the zeros of its related quadratic function to sketch parabola graphs. The lesson also introduces the concept of a double root, found when a perfect square trinomial factors to a repeated linear factor.

Section 1

Zero Product Property

Property

Let $a$ and $b$ be real numbers. If $ab = 0$ , then $a = 0$ or $b = 0$ .

Examples

Section 2

Square Root Property

Property

If $x^2 = a$ , then $x = \pm\sqrt{a}$ for any $a > 0$ .

Examples

Section 3

Solving Quadratic Equations by Factoring out the GCF

Property

For a quadratic equation where the constant term is zero, such as

ax^2 + bx = 0

, factor out the Greatest Common Factor (GCF) to rewrite the equation as:

cx(dx + e) = 0

where $cx$ is the GCF of the terms $ax^2$ and $bx$ . Then apply the Zero Product Property.

Examples

Solve $x^2 - 7x = 0$ : Factor out the GCF $x$ to get $x(x - 7) = 0$ . Setting each factor to zero gives $x = 0$ or $x = 7$ .
Solve $6x^2 + 18x = 0$ : Factor out the GCF $6x$ to get $6x(x + 3) = 0$ . Setting each factor to zero yields $6x = 0 \implies x = 0$ or $x + 3 = 0 \implies x = -3$ .

Explanation

When solving a quadratic equation that lacks a constant term, every term will contain the variable $x$ . The most efficient first step is to factor out the Greatest Common Factor (GCF), which includes $x$ and any shared numerical values. Once the expression is written as a product of factors equal to zero, you can easily apply the Zero Product Property. By setting each individual factor to zero, you will find the solutions, noting that one of the roots will always be zero.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Zero Product Property

Property

Let $a$ and $b$ be real numbers. If $ab = 0$ , then $a = 0$ or $b = 0$ .

Examples

Section 2

Square Root Property

Property

If $x^2 = a$ , then $x = \pm\sqrt{a}$ for any $a > 0$ .

Examples

Section 3

Solving Quadratic Equations by Factoring out the GCF

Property

For a quadratic equation where the constant term is zero, such as

ax^2 + bx = 0

, factor out the Greatest Common Factor (GCF) to rewrite the equation as:

cx(dx + e) = 0

where $cx$ is the GCF of the terms $ax^2$ and $bx$ . Then apply the Zero Product Property.

Examples

Solve $x^2 - 7x = 0$ : Factor out the GCF $x$ to get $x(x - 7) = 0$ . Setting each factor to zero gives $x = 0$ or $x = 7$ .
Solve $6x^2 + 18x = 0$ : Factor out the GCF $6x$ to get $6x(x + 3) = 0$ . Setting each factor to zero yields $6x = 0 \implies x = 0$ or $x + 3 = 0 \implies x = -3$ .

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Zero Product Property

Property

Let $a$ and $b$ be real numbers. If $ab = 0$ , then $a = 0$ or $b = 0$ .

Examples

Section 2

Square Root Property

Property

If $x^2 = a$ , then $x = \pm\sqrt{a}$ for any $a > 0$ .

Examples

Section 3

Solving Quadratic Equations by Factoring out the GCF

Property

For a quadratic equation where the constant term is zero, such as

ax^2 + bx = 0

, factor out the Greatest Common Factor (GCF) to rewrite the equation as:

cx(dx + e) = 0

where $cx$ is the GCF of the terms $ax^2$ and $bx$ . Then apply the Zero Product Property.

Examples

Solve $x^2 - 7x = 0$ : Factor out the GCF $x$ to get $x(x - 7) = 0$ . Setting each factor to zero gives $x = 0$ or $x = 7$ .
Solve $6x^2 + 18x = 0$ : Factor out the GCF $6x$ to get $6x(x + 3) = 0$ . Setting each factor to zero yields $6x = 0 \implies x = 0$ or $x + 3 = 0 \implies x = -3$ .