1. Factorisation. Single Brackets.
Multiply out the bracket below:
2x ( 4x – 6 )
Example 2
Factorise: 40x2 – 5x
= 8x2
- 12x = 5
( 8x2
- x )
Factorisation is the reversal of the above process. That is to say we put the brackets back in.
= 5 x
( 8 x
- 1 )
Example 1
Factorise: 4x2 – 12 x
Hint:Numbers First.
= 4
( x2 – 3x)
Hint:Now Letters
= 4x
( x – 3 )

2. What Goes In The Box ? Factorise fully : 12 x2 – 6 x
Now factorise the following:
(1) 14 x x
=7x( x + 1) 6
( 2x2
- x )
(2) 4x – 12 x 2
= 4x ( 1 – 3x) 6x
( 2x
- 1 )
(3) 6ab – 2ad
=2a( 3b – d)
(4) 12 a2 b – 6 a b2
= 6ab ( a – b)

3. A Difference Of Two Squares.
Consider what happens when you multiply out :
( x + y ) ( x – y)
Now you try the example below:
Example.
Multiply out:
( 5 x + 7 y )( 5 x – 7 y )
= x
( x – y )
+ y
( x – y )
- y 2
=x 2
- xy
+ xy
Answer:
= x2
- y2
= 25 x 2
- 49 y 2
This is a difference of two squares.

4. What Goes In The Box ? Mutiply out: (1) ( 3 x + 6 y ) ( 3 x – 6 y)

5. Factorising A Difference Of Two Squares.
By considering the brackets required to produce the following factorise the following examples directly:
Examples
(5) 4x
(1) x
= ( 2x
- 6 )
( 2x
+ 6 )
= ( x
- 3 )
( x
+ 3 )
(2) x
(6) 9x y 2
= ( x
- 4 )
( x
+ 4 )
= ( 3x
- 4y )
( 3x
+ 4y )
(3) x
(7) 100g k 2
= ( 10
g – 7k )
( 10g
+ 7k )
= ( x
- 5 )
( x
+ 5 )
(8) 144d w 2
(4) x 2 - y 2
= ( 12d
- 6 w)
( 12d
+ 6w )
= ( x
- y )
( x
+ y )

6. What Goes In The Box ? Multiply out the brackets below:
(3x – 4 ) ( 2x + 7) 3x
(2x + 7) -4
(2x + 7)
6x 2
-8x
-28
+21x
You are now about to discover how to put the double brackets back in.
6x 2
-28
+13x

7. Factorising A Quadratic.
Follow the steps below to put a double bracket back into a quadratic equation.
Process.
Step 1:
Consider the factors of the coefficient in front of the x and the constant.
Factorise the quadratic:
x2 – 2x - 15
Factors 5x 1 15
Step 2 :
Create the x coefficient from two pairs of factors.
= (x 5) ( x 3) 1 1 1 15 3x 3 5
Step 3
Place the four numbers in the pair of brackets looking at outer and inner pairs to determine the signs.
3x – 5x = - 2x
x coefficient = 2
(1 x 5) – (1 x 3 ) = 2
= (x - 5) ( x +3)

8. More Quadratic Factorisation Examples.
Factorise the quadratic:
x2 + 3x - 10
Factors 1 10 1 1 1 10 5x
= (x 5) ( x 2) 2 5 2x
x coefficient = 3
(1 x 5) - (1 x 2 ) = 3
= (x + 5) ( x - 2 )
Signs in brackets.
5x – 2x = 3x

9. Quadratic Factorisation Example 2
Factorise the quadratic:
x2 – 8x + 12
Factors 1 12 1 1 1 12 6x
= (x 6) ( x 2) 2 6 2x 3 4
x coefficient = 8
= (x - 6) ( x -2 )
(1 x 6) + (1 x 2 ) = 8
Signs in brackets.
- 6x – 2x = - 8x

10. Quadratic Factorisation Example 3.
Factorise the quadratic:
6 x x – 10
Factors 6 10 1 6 1 10 4x
= (3x 2) ( 2x 5) 2 3 2 5
15x
x coefficient = 11
= ( 3x - 2) ( 2 x + 5)
(3 x 5) – (2 x 2 ) = 11
Numbers together.
Signs in brackets.
Numbers apart.
15 x – 4x = 11x

11. Quadratic Factorisation Example 4
Factorise the quadratic:
10 x x – 28
Factors 10 28 1 10 1 28 8x 2 5 2 14
= (5x 4) ( 2x 7) 4 7
35x
x coefficient = 27
= ( 5x - 4) ( 2 x + 7)
(5 x 7) – (2 x 4 ) = 27
Signs in brackets.
35 x – 8x = 27x

12. What Goes In The Box ? 6 x2 – x – 2 Factorise the quadratic: Factors 61 6 1 2 2 3
= (3x 2) ( 2x 1)
x coefficient -1
= ( 3x - 2) ( 2 x + 1)
(2 x 2) – (1 x 3 ) = 1
Signs in brackets.
3 x – 4x = -x

13. What Goes In The Box 2 15 x2 – 19x + 6 Factorise the quadratic:
Factors 15 6 1 15 1 6 3 5 2 3
= (3x 2) ( 5x 3)
x coefficient
-19
= ( 3x - 2) ( 5 x - 3)
(3 x 3) + (5 x 2 ) = 19
Signs in brackets.
- 9 x – 10x = - 19x