Factorising quadratics

Foundation and Higher

The idea

To factorise $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$ . For $x^2 + 2x - 15$ they are $5$ and $-3$ , so it is $(x + 5)(x - 3)$ . A difference of two squares: $x^2 - 49 = (x + 7)(x - 7)$ .

List factor pairs of $c$ , with signs.
Pick the pair that adds to $b$ .
Expand to check.

Watch out Expand your brackets back out: the middle term catches most sign slips.

See it worked

Worked exampleNon-calculator4 marks

Factorise each expression.

(a) Factorise $x^2 + 7x + 12$

(b) Factorise $x^2 - 5x + 6$

(a) Find two numbers that multiply to 12 and add to 7: $3$ and $4$ .
$x^2 + 7x + 12 = (x + 3)(x + 4)$
(b) The numbers multiply to $+6$ and add to $-5$ , so both are negative: $-2$ and $-3$ .
$x^2 - 5x + 6 = (x - 2)(x - 3)$

Answer (a) $(x + 3)(x + 4)$ ; (b) $(x - 2)(x - 3)$

Your turn

Question 1Non-calculator3 marks

Factorise each expression.

(a) Factorise $y^2 + 10y + 21$

(b) Factorise $p^2 - 9$

Question 2Non-calculator4 marks

Consider $x^2 - 2x - 24$ .

(a) Factorise $x^2 - 2x - 24$ .

(b) Hence solve $x^2 - 2x - 24 = 0$ .

Question 3Non-calculator5 marks

Factorise each expression fully.

(a) Factorise $x^2 + 2x - 15$

(b) Factorise $x^2 - 49$

In past papers

63 official questions in 59 papers since 2017. The latest:

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