GCSE Factorising revision
A quick, oneoff 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 (lettersquared and letter)

you can divide them both by $x$
 $x \times x = x^2$

$x \times 4 = 4x$
 $x^2  4x = x(x4)$
Try these:
$b^2 + 9b$ $c^2  2c$ $d^2 + 10d$
(3) Common number and letter
$6x^2  4x$
 two terms (lettersquared and letter)

you can divide them both by $2x$
 $2x \times 3x = 6x^2$

$2x \times 2 = 4x$
 $6x^2  4x = 2x(3x2)$
Try these:
$2x^2 + 8x$ $3x^2  6x$ $4x^2  10x$
(4) Difference of two squares
$4x^2  25$
 two terms (both squares)

Use $(a+b)(ab) = a^2  b^2$
 $4x^2 = (2x)^2$, 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 (lettersquared, 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.
 $x^2  2x  15 = (x+3)(x5)$
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 (lettersquared, 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 20160508 to correct wrong letters in last two questions. Thanks, Rosie, for pointing out my error.