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.