How do you solve a quadratic equation by factoring?

Get 0 on one side, factor the quadratic into two brackets, set each bracket equal to 0, and solve each one.

How do you factor x² + bx + c?

Find two numbers that multiply to c and add to b. If they are m and n, it factors to (x + m)(x + n).

Why must the equation equal 0 before you factor?

The zero product property says a product is 0 only if a factor is 0. If the product equals any other number, a bracket does not have to equal that number.

What if a quadratic will not factor?

Use the quadratic formula, which works for every quadratic, even when the solutions are not whole numbers.

Part 29 of 30 · QuadraticsSee the whole course

How to solve a quadratic by factoring

By The Pip team · October 24, 2026 · 8 min read

Short answer

Move everything to one side so it equals 0. Factor the quadratic into two brackets. Then set each bracket equal to 0 and solve. For x² − 5x + 6 = 0 it factors to (x − 2)(x − 3) = 0, so x = 2 or x = 3.

The method in four steps

Get 0 on one side. Move every term over so the equation reads ax² + bx + c = 0.
Find the pair. For x² + bx + c, find two numbers that multiply to c and add to b.
Factor. If the pair is m and n, then x² + bx + c = (x + m)(x + n).
Zero product. A product is zero only when a bracket is zero (see roots from the factored form). Set each bracket to 0.

Example 1: x² − 5x + 6 = 0

Find two numbers that multiply to 6 and add to −5: −2 and −3.
So (x − 2)(x − 3) = 0.
x − 2 = 0 gives x = 2, and x − 3 = 0 gives x = 3.
Check: 2² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓ and 3² − 15 + 6 = 0 ✓.

Example 2: x² + 2x − 8 = 0

Multiply to −8, add to 2: 4 and −2.
(x + 4)(x − 2) = 0, so x = −4 or x = 2.

Example 3: x² = 3x + 10, which needs a first step

It is not equal to 0 yet. Subtract 3x and 10 from both sides: x² − 3x − 10 = 0.
Multiply to −10, add to −3: −5 and 2.
(x − 5)(x + 2) = 0, so x = 5 or x = −2.

Step through these and two more in the figure below. The parabola on the graph is the one-sided form, and the solutions are where it lands on the x-axis.

x² − 5x + 6 = 0

x² − 5x + 6 = 0
already 0 on one side
multiply to 6, add to −5: −2 and −3
(x − 2)(x − 3) = 0
x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3

Start with x² − 5x + 6 = 0. The first job is to get 0 on one side.

Two shortcuts

A difference of squares. x² − 9 = 0 factors to (x − 3)(x + 3) = 0, so x = 3 or x = −3.
A common factor. 2x² − 8x = 0 has no constant. Factor out 2x: 2x(x − 4) = 0, so x = 0 or x = 4.

When it will not factor

Many quadratics do not factor into whole numbers, like x² − 4x + 1 = 0. For those you need the formula in the next post.

Watch it explained in under a minute

The same idea, narrated step by step. Square version for sharing: download the 1:1 video.

The four mistakes to avoid

Not getting 0 on one side first. The zero product rule only works when the product equals 0.
Getting the signs of the pair wrong. They must add to b, the number with the sign it has. In x² − 5x + 6 the pair is −2 and −3, not 2 and 3.
Dividing both sides by x and losing a solution. For x² = 4x, dividing by x loses x = 0. Move it across and factor instead.
Stopping after one bracket. Each bracket gives its own solution.

Try it yourself

Tap a question to see the answer.

1. x² − 7x + 12 = 0

Multiply to 12, add to −7: −3 and −4. (x − 3)(x − 4) = 0, so x = 3 or x = 4.

2. x² + x − 12 = 0

Multiply to −12, add to 1: 4 and −3. (x + 4)(x − 3) = 0, so x = −4 or x = 3.

3. x² = 4x

Subtract 4x: x² − 4x = 0. Factor out x: x(x − 4) = 0, so x = 0 or x = 4.

Where this fits in

Factoring is quick when the numbers are friendly. When they are not, one formula solves every quadratic: the quadratic formula and the discriminant.