Solving Quadratic Equations by Factoring: A Practical Guide

A friendly step-by-step lesson for students, teachers, homeschool families, and curious algebra learners.

Big idea: Rewrite a quadratic as a product, use the Zero Product Property, and solve each factor.

What Is a Quadratic Equation?

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b, and c are numbers and a ≠ 0. The highest power of the variable is 2.

Factoring is one method for solving a quadratic when the expression can be written as two simpler factors. It is often quickest when the numbers factor cleanly.

The Rule You Need

If A × B = 0, then A = 0 or B = 0

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This is the Zero Product Property. A product can equal zero only when at least one factor equals zero. It is the bridge between a factored quadratic and its solutions.

The Four-Step Method

1. Set the equation equal to zero. Move all terms to one side if necessary.
2. Factor the quadratic completely. Look for a greatest common factor first, then factor the trinomial or use a special pattern.
3. Set each factor equal to zero. Apply the Zero Product Property.
4. Solve and check. Substitute each answer into the original equation.

Worked Example 1: A Monic Quadratic

Solve: x² + 5x + 6 = 0

Find two numbers whose product is 6 and whose sum is 5: 2 and 3.

(x + 2)(x + 3) = 0

Set each factor equal to zero:

x + 2 = 0, so x = −2
x + 3 = 0, so x = −3

Solutions: x = −2 and x = −3

Worked Example 2: A Leading Coefficient Greater Than 1

Solve: 2x² + 7x + 3 = 0

We need factors of 2 × 3 = 6 that add to 7: 6 and 1. Split the middle term:

2x² + 6x + x + 3 = 0

Group and factor:

2x(x + 3) + 1(x + 3) = 0
(2x + 1)(x + 3) = 0

Therefore, 2x + 1 = 0 gives x = −1 ÷ 2, and x + 3 = 0 gives x = −3.

Solutions: x = −1 ÷ 2 and x = −3

Worked Example 3: A Difference of Squares

Solve: x² − 25 = 0

Recognize the difference of squares: a² − b² = (a − b)(a + b).

(x − 5)(x + 5) = 0

So x − 5 = 0 gives x = 5, and x + 5 = 0 gives x = −5.

Solutions: x = 5 and x = −5

A Memory Tip

Zero first, factor second, split third, check last.
In short: Set → Factor → Zero → Check.

Common Mistakes to Avoid

• Not setting the equation equal to zero: The Zero Product Property requires a product equal to zero.
• Forgetting both solutions: Set every factor equal to zero.
• Sign errors: If x − 4 = 0, then x = 4; if x + 4 = 0, then x = −4.
• Stopping before factoring completely: Keep looking for a common factor or special pattern.
• Skipping the check: Substitute solutions into the original equation, not only the factored form.

Practice Questions

Try each problem before reading the answers. Show all four steps.

Question	Solve
1	x2 + 7x + 12 = 0
2	x2 − x − 12 = 0
3	3x2 + 10x + 8 = 0
4	4x2 − 9 = 0

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Answers and Quick Checks

1. (x + 3)(x + 4) = 0, so x = −3 or x = −4.

2. (x − 4)(x + 3) = 0, so x = 4 or x = −3.

3. (3x + 4)(x + 2) = 0, so x = −4 ÷ 3 or x = −2.

4. (2x − 3)(2x + 3) = 0, so x = 3 ÷ 2 or x = −3 ÷ 2.

When Factoring Is Not the Best Choice

Factoring is excellent when the factors are easy to find. If a quadratic does not factor neatly over the integers, the quadratic formula or completing the square may be more efficient. A strong algebra solver chooses the method that fits the equation.

Final Takeaway

To solve a quadratic by factoring, make zero, factor completely, set each factor equal to zero, and check your answers.

Set → Factor → Zero → Check