How to Factor Quadratics Step by Step

Factoring turns a quadratic into two brackets you can solve in seconds. Learn the method, see worked examples, and avoid the mistakes most students make.

By Muddassar Ali · Oct 27, 2026 · 5 min read

Factoring turns a quadratic equation into two brackets you can solve in seconds — when the numbers cooperate. It is the fastest of the four standard methods for solving quadratics, and the one most students meet first.

What factoring means

A quadratic has the form $a x^{2} + b x + c = 0$ . To factor it means to rewrite it as two brackets multiplied together, like this:

$x^{2} - 5 x + 6 = 0 \Rightarrow (x - 2) (x - 3) = 0$

Once the equation is in brackets, each bracket gives one answer: $x = 2$ or $x = 3$ . That step works because if two things multiply to zero, at least one of them must be zero.

The method, step by step

Take $x^{2} - 5 x + 6 = 0$ as the example, where $a = 1$ , $b = - 5$ , $c = 6$ :

Find two numbers that multiply to $c$ and add to $b$ . Here you need two numbers that multiply to $6$ and add to $- 5$ : $- 2$ and $- 3$ .
Write the brackets. Those two numbers go into the brackets: $(x - 2) (x - 3)$ .
Set each bracket to zero. $x - 2 = 0$ gives $x = 2$ ; $x - 3 = 0$ gives $x = 3$ .
Check by expanding. $(x - 2) (x - 3) = x^{2} - 5 x + 6$ . It matches, so the answers are right.

When $a$ is not $1$ , the same idea applies but the search is longer, because the numbers must also account for $a$ . In that case, or when the numbers are not small integers, the quadratic formula always works instead.

Two more worked examples

Example 1: $x^{2} + 7 x + 12 = 0$ . Two numbers that multiply to $12$ and add to $7$ : $3$ and $4$ . So $(x + 3) (x + 4) = 0$ , giving $x = - 3$ or $x = - 4$ .

Example 2: $x^{2} - 9 = 0$ . This is a difference of two squares: $(x - 3) (x + 3) = 0$ , giving $x = 3$ or $x = - 3$ . Note the $\pm$ — forgetting the negative root is the most common slip here.

Common mistakes

Forgetting a sign. If $c$ is positive, both numbers have the same sign as $b$ ; if $c$ is negative, the numbers have opposite signs.
Not moving everything to one side first. Factoring only works when the equation equals zero. Move every term across before you start.
Stopping at the brackets. The brackets are not the answer — set each one to zero and solve.

When NOT to factor

Factoring is the wrong tool when the roots are not small integers — for example, $x^{2} + 6 x + 2 = 0$ does not factor neatly. That is when you switch to completing the square or the quadratic formula. Choosing the right method is covered in How to Solve Quadratic Equations: 4 Methods Explained.

Practice it

Want to check your work? Try the Factoring Calculator — every answer is verified by substitution. Then try similar problems on the practice page until the number search feels automatic.

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