How to Factorise Quadratics: Sum-Product Method and Difference of Two Squares

Q: How do you factorise x² + bx + c?

Find two numbers that multiply to c and add to b. If they are p and q, the answer is (x + p)(x + q).

Q: What if a quadratic will not factorise?

Some quadratics have no neat factors. Use completing the square or the quadratic formula instead; they work for every quadratic.

Factorising as expanding in reverse, the sum-and-product method, how the signs of $b$ and $c$ guide you, the difference of two squares, and how to handle a leading coefficient that is not 1.

3 min read Supports 4 lessons

Expanding turns brackets into a sum: $(x + 3)(x + 4) = x^2 + 7x + 12$ . Factorising goes the other way, from the sum back to the brackets. It is like being told $12$ and asked which numbers multiply to make it.

Always look for a common factor first

Before anything else, pull out the greatest common factor of every term.

Worked example

Factorise $12x^3 - 18x^2$ .

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The GCF of the numbers is $6$ ; of $x^3$ and $x^2$ it is $x^2$ .
$6x^2$

The sum-and-product method

Expand $(x + p)(x + q)$ and look at where each coefficient comes from:

(x + p)(x + q) = x^2 + (p + q)x + pq

Key idea

To factorise $x^2 + bx + c$ , find two numbers $p$ and $q$ that add to $b$ and multiply to $c$ .

$x^2 + 7x + 12$
Need two numbers: product 12, sum 7.

1/4

Using the signs

Tip

If $c > 0$ , both numbers have the same sign as $b$ . If $c < 0$ , the numbers have opposite signs, and the bigger one has the sign of $b$ .

Worked example

Factorise $x^2 - 2x - 15$ .

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The product is negative, so the signs differ. The sum is negative, so the bigger number is negative.

Difference of two squares

When a quadratic has no middle term and is one square minus another, it factorises instantly.

Difference of two squares

a^2 - b^2 = (a - b)(a + b)

e.g. $x^2 - 49 = (x - 7)(x + 7)$

The middle terms cancel when you expand: $(a - b)(a + b) = a^2 + ab - ab - b^2$ . A grid shows why: the two cross terms are equal and opposite.

Watch out

A sum of two squares, like $x^2 + 9$ , does not factorise using real numbers.

When the leading coefficient is not 1

For $ax^2 + bx + c$ , find two numbers that multiply to $ac$ and add to $b$ . Use them to split the middle term, then factorise in pairs.

Worked example

Factorise $2x^2 + 7x + 3$ .

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$ac = 6$ . Two numbers with product $6$ and sum $7$ : $6$ and $1$ .

Key idea

Factorising is how quadratic equations are solved: once $x^2 + 7x + 12 = (x + 3)(x + 4) = 0$ , one bracket must be zero, so $x = -3$ or $x = -4$ .

Not every quadratic factorises neatly. If you cannot find the numbers, the quadratic formula always works.

Common questions

How do you factorise x² + bx + c?: Find two numbers that multiply to $c$ and add to $b$ . If they are $p$ and $q$ , the answer is $(x + p)(x + q)$ .
What if a quadratic will not factorise?: Some quadratics have no neat factors. Use completing the square or the quadratic formula instead; they work for every quadratic.

Practise factorising quadratics

The Factoring Quadratics lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

How do you factorise x² + bx + c?

Find two numbers that multiply to $c$ and add to $b$ . If they are $p$ and $q$ , the answer is $(x + p)(x + q)$ .

What if a quadratic will not factorise?

Some quadratics have no neat factors. Use completing the square or the quadratic formula instead; they work for every quadratic.