8.4Solving Quadratics by Factoring
Use the zero product property to turn a factored quadratic into two simple linear equations — and find the roots, zeros, and x-intercepts of any factorable parabola.
Why This Matters
Solving quadratics by factoring is the fastest method when it applies — and it reinforces your factoring skills from Unit 7. You'll use this technique in Algebra 2, Precalculus, and whenever you need to find the zeros of a polynomial.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does the zero product property allow us to use factoring to find the solutions of a quadratic equation?
Lesson Overview
Factoring is the process of rewriting a quadratic expression as a product of two binomials. Once factored, the zero product property lets you split the equation into two simple linear equations and solve each one. The solutions are called roots or zeros — they are also the x-intercepts of the parabola. The key steps are always the same: write in standard form, factor completely, set each factor equal to zero, and solve.
Worked Examples
Solve x² + 5x + 6 = 0
Already in standard form. a=1.
Find two numbers: multiply to 6, add to 5 → 2 and 3
Factor: (x+2)(x+3) = 0
Zero product: x+2=0 or x+3=0
Solutions: x=−2 or x=−3
Solve x² − 7x + 12 = 0
Find two numbers: multiply to 12, add to −7 → −3 and −4
Factor: (x−3)(x−4) = 0
Zero product: x=3 or x=4
Solve 2x² + 8x = 0
Standard form: already equals 0.
Factor out GCF: 2x(x+4) = 0
Zero product: 2x=0 or x+4=0
Solutions: x=0 or x=−4
Solve x² − 16 = 0
Difference of squares: a²−b² = (a+b)(a−b)
x²−16 = (x+4)(x−4) = 0
Zero product: x=−4 or x=4
Solve 2x² + 7x + 3 = 0 using the AC method
a=2, b=7, c=3 → ac=6
Two numbers: multiply to 6, add to 7 → 1 and 6
Split: 2x²+x+6x+3
Group: x(2x+1)+3(2x+1) = (x+3)(2x+1) = 0
Zero product: x=−3 or x=−1/2
Guided Practice
Guided Practice Video: Solving by Factoring
Review solving quadratic equations by factoring and applying the zero product property before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Solve x² + 7x + 10 = 0
Hint: Find two numbers that multiply to 10 and add to 7.
Solve x² − 9x + 18 = 0
Hint: Find two numbers that multiply to 18 and add to −9. Both will be negative.
Solve 3x² − 12x = 0
Hint: Factor out the GCF first. What do 3x² and −12x have in common?
Solve x² − 25 = 0
Hint: Recognize the difference of squares pattern: a²−b²=(a+b)(a−b).
Solve 3x² + 10x + 3 = 0 using the AC method
Hint: ac = 9. Find two numbers that multiply to 9 and add to 10: 1 and 9.
Key Vocabulary
Zero Product Property
If A·B=0, then A=0 or B=0. The foundation of solving by factoring.
Root / Zero
A value of x that makes the equation equal to zero. Also the x-coordinate of an x-intercept.
GCF (Greatest Common Factor)
The largest factor shared by all terms. Always factor out the GCF first.
Trinomial
A polynomial with three terms. Standard form ax²+bx+c is a trinomial.
AC Method
A factoring strategy for ax²+bx+c (a≠1): find two numbers that multiply to ac and add to b, then split the middle term.
Difference of Squares
a²−b² = (a+b)(a−b). Applies when a quadratic has no middle term and both terms are perfect squares.
Practice Questions
Interactive Practice — 5 Questions
What are the solutions to (x − 3)(x + 5) = 0?
Factor x² + 7x + 12.
Solve x² − 9 = 0 by factoring.
What is the first step to solve 2x² + 6x = 0?
Solve x² − 5x + 6 = 0.
Independent Practice
Independent Practice
Solve: x² + 6x + 8 = 0
Solve: x² − 4x − 12 = 0
Solve: x² − 8x + 15 = 0
Solve: 4x² − 16x = 0
Solve: x² − 49 = 0
Common Mistakes
Forgetting to set the equation equal to zero before factoring — e.g., factoring x² + 5x = 6 directly.
Move all terms to one side first: x² + 5x − 6 = 0. Then factor.
Setting the product equal to 6 instead of 0 — e.g., from (x + 6)(x − 1) = 6, writing x + 6 = 6 or x − 1 = 6.
The zero product property only works when the product equals zero. Set each factor equal to 0.
Finding factors that multiply to c but don't add to b.
You need two numbers that both multiply to c AND add to b. Check both conditions.
Forgetting to check the GCF before factoring the trinomial.
Always factor out the GCF first. It simplifies the remaining trinomial and prevents errors.
Math Tips
Always move all terms to one side first so the equation equals zero before factoring.
Zero Product Property: if A·B = 0, then A = 0 or B = 0. This only works when one side is zero.
Check your solutions by substituting each one back into the original equation.
SAT tip: factoring is the fastest method when the quadratic factors nicely — always try it first.