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:

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

(5) Regular quadratic

$x^2 - 2x - 15$

  • 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.

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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

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