A3AlgebraFoundation & Higher

Factorising

Factorising is the reverse of expanding — you put the brackets back in. It is one of the most important algebra skills because it is the key to solving quadratic equations.

40 min Video by Maths Genie AQA GCSE Maths

Factorising Single BracketsWatch the walkthrough, then read the notes below.

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What you'll learn

Factorise by taking out a common factor
Factorise quadratics of the form x² + bx + c
Use the difference of two squares
Check by expanding

Common factors

Find the highest factor shared by every term and write it outside a bracket: $6 x + 9 = 3 (2 x + 3)$ . This can include letters too: $x^{2} + x = x (x + 1)$ .

Quadratics and difference of two squares

To factorise $x^{2} + b x + c$ , find two numbers that multiply to $c$ and add to $b$ . The $difference of two squares$ is a special pattern: $a^{2} - b^{2} = (a + b) (a - b)$ .

x^{2} + b x + c = (x + p) (x + q), pq = c, p + q = b

Find two numbers: multiply to c, add to b.

1Two numbers multiply to 6 and add to 5: that’s 2 and 3.

2So it factorises into

(x + 2) (x + 3)

Answer

(x + 2) (x + 3)

Tip — Always check by expanding your brackets back out.

Remember these

ab + a c = a (b + c)

Common factor.

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

Difference of two squares.

Watch out for these

Not taking out the highest common factor:

6 x + 9 = 3 (2 x + 3)

✓ but

6 x + 9 = 6 (x + 1.5)

✗.

Take out the highest factor common to every term.

Getting the signs wrong in the brackets.

Check the two numbers really multiply to c and add to b.

Key takeaways

Common factor: take out the highest shared factor.
x² + bx + c: two numbers multiply to c, add to b.
a² − b² = (a+b)(a−b).

Test yourself

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