GCSE Factorising revision

A quick, one-off masterclass in how to put things into brackets today - six methods of factorising you need to know to do well at GCSE maths.

(1) Common number

$3a + 6$

• two terms (letter and number, no squares)
• you can divide them both by 3
• $3 \times a = 3a$
• $3 \times 2 = 6$
• $3a + b = 3(a + 2)$

Try these:

$9b + 12$
$12c - 8$
$90d + 15$
$7e - 14$

(2) Common letter

$x^2 - 4x$

• two terms (letter-squared and letter)
• you can divide them both by $x$
• $x \times x = x^2$
• $x \times -4 = -4x$
• $x^2 - 4x = x(x-4)$

Try these:

$b^2 + 9b$
$c^2 - 2c$
$d^2 + 10d$

(3) Common number and letter

$6x^2 - 4x$

• two terms (letter-squared and letter)
• you can divide them both by $2x$
• $2x \times 3x = 6x^2$
• $2x \times -2 = -4x$
• $6x^2 - 4x = 2x(3x-2)$

Try these:

$2x^2 + 8x$
$3x^2 - 6x$
$4x^2 - 10x$

(4) Difference of two squares

$4x^2 - 25$

• two terms (both squares)
• Use $(a+b)(a-b) = a^2 - b^2$
• $4x^2 = (2x)^2$, so $a = 2x$
• $25 = 5^2$, so $b = 5$
• $4x^2 - 25 = (2x -5)(2x + 5)$

Try these:

$x^2 - 9$
$9x^2 - 1$
$4a^2 - 9b^2$

$x^2 - 2x - 15$

• Three terms (letter-squared, letter and number)
• Look for two numbers $p$ and $q$ such that:
• $pq = -15$; and
• $p+q = -2$. (Adding things in the middle would be the end of the Times!)
• Not too many possibilities that multiply to -15: 1 and -15, 3 and -5, 5 and -3, 15 and -1. Only 3 and -5 work.
• $x^2 - 2x - 15 = (x+3)(x-5)$

Try these:

$x^2 + 3x + 2$
$x^2 - 3x + 2$
$x^2 + 13x + 36$
$x^2 + 2x - 35$

(6) Quadratic with a number in front

$4x^2 + 8x + 3$

• Three terms (letter-squared, letter and number)
• More difficult! Magic number is $4 \times 3 = 12$
• Want two numbers $p$ and $q$ such that:
• $pq = 12$
• $p + q = 8$
• 2 and 6 work!
• Split up $8x$ as $2x + 6x$ and write out: $4x^2 + 2x + 6x + 3$
• Factorise first half: $2x(2x + 1)$
• Factorise second half: $3(2x + 1)$
• Combine: $(2x+3)(2x+1)$ - phew!

Try these:

* $2x^2 + 3x + 1$
* $3y^2 + 8y - 3$
* $4z^2 + 5z + 1$

* Edited 2016-05-08 to correct wrong letters in last two questions. Thanks, Rosie, for pointing out my error.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

3 comments on “GCSE Factorising revision”

• Rosie

how would you factorise the very last two questions of (6) when the letters are different?

• Colin

Yikes! You would sneakily edit the quiz to correct the questions! Good spot, thank you. Fixed now.

• Naomi Boitumelo

Dear Colin
thanks very much for your help,i do understand better than before.keep on posting

This site uses Akismet to reduce spam. Learn how your comment data is processed.