Quadratic-Factorisation.-Creditnbcbv.ppt

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

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

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

4. What Goes In The Box ?
(1) ( 3 x + 6 y ) ( 3 x – 6 y)
(2) ( 2 x – 4 y ) ( 2 x + 4 y)
(3) ( 8 x + 9 y ) ( 8 x – 9 y)
(3) ( 5 x – 7 y ) ( 5 x + 7 y)
(4) ( x – 11 y ) ( x + 11 y)
(5) ( 7 x + 2 y ) ( 7 x – 2 y)
(6) ( 5 x – 9 y ) ( 5 x + 9 y)
(7) ( 3 x + 9 y ) ( 3 x – 9 y)
= 9 x
2
– 36 y
2
= 4 x
2
– 16 y
2
= 64 x
2
– 81 y
2
= 25 x
2
– 49 y
2
= x
2
– 121 y
2
= 49 x
2
– 4 y
2
= 25 x
2– 81 y
2
= 9 x
2
– 81 y
2
Mutiply out:

5. Factorising A Difference Of Two Squares.
By considering the brackets required to produce the following
factorise the following examples directly:
Examples
(1) x
2
- 9
(2) x
2
- 16
(3) x
2
- 25
(4) x
2
- y
2
(5) 4x
2
- 36
(6) 9x
2
- 16y
2
(7) 100g
2
- 49k
2
(8) 144d
2
- 36w
2
( x - 3 )= ( x + 3 )
= ( x - 4 )( x+ 4 )
= ( x - 5 )( x+ 5 )
= ( x - y )( x+ y )
= ( 2x - 6 )( 2x+ 6 )
= ( 3x - 4y )( 3x+ 4y )
= ( 10g – 7k )( 10g+ 7k )
= ( 12d - 6 w)( 12d+ 6w )

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

7. Factorising A Quadratic.
Follow the steps below to put a double bracket back into a
quadratic equation.
Factorise the quadratic:
x
2
– 2x - 15
Process.Step 1:
Consider the factors of the
coefficient in front of the x
and the constant.
Factors
1 15
11115
35
Step 2 :
Create the x coefficient from
two pairs of factors.
x coefficient = 2
(1 x 5) – (1 x 3 ) = 2
Step 3
Place the four numbers in the
pair of brackets looking at
outer and inner pairs to
determine the signs.
= (x 5) ( x 3)
5x
3x
3x – 5x = - 2x
= (x - 5) ( x +3)

8. More Quadratic Factorisation Examples.
Example 1.
Factorise the quadratic:
x
2
+ 3x - 10
Factors
1 10
11110
25
x coefficient = 3
(1 x 5) - (1 x 2 ) = 3
= (x 5) ( x 2)
5x
2x
Signs in brackets.
= (x + 5) ( x - 2 )
5x – 2x = 3x

9. Quadratic Factorisation Example 2
Factorise the quadratic:
x
2
– 8x + 12
Factors
1 12
11112
3
6
x coefficient = 8
= (x 6) ( x 2)
6x
2x
Signs in brackets.
= (x - 6) ( x -2 )
- 6x – 2x = - 8x
(1 x 6) + (1 x 2 ) = 8
2
4

10. Quadratic Factorisation Example 3.
Factorise the quadratic:
6 x
2
+ 11x – 10
Factors
6 10
16110
2325
x coefficient = 11
(3 x 5) – (2 x 2 ) = 11
= (3x 2) ( 2x 5)
4x
15x
Signs in brackets.
15 x – 4x = 11x
= ( 3x - 2) ( 2 x + 5)
Numbers together.
Numbers apart.

11. Quadratic Factorisation Example 4
Factorise the quadratic:
10 x
2
+ 27x – 28
Factors
10 28
110128
25214
x coefficient = 27
(5 x 7) – (2 x 4 ) = 27
= (5x 4) ( 2x 7)
8x
35x
Signs in brackets.
35 x – 8x = 27x
= ( 5x - 4) ( 2 x + 7)
47

12. What Goes In The Box ?
Factorise the quadratic:
6 x
2
– x – 2
Factors
6 2
1612
23
x coefficient
(2 x 2) – (1 x 3 ) = 1
= (3x 2) ( 2x 1)
Signs in brackets.
3 x – 4x = -x
= ( 3x - 2) ( 2 x + 1)
-1

13. What Goes In The Box 2
Factorise the quadratic:
15 x
2
– 19x + 6
Factors
15 6
11516
35
x coefficient
(3 x 3) + (5 x 2 ) = 19
= (3x 2) ( 5x 3)
Signs in brackets.
- 9 x – 10x = - 19x
= ( 3x - 2) ( 5 x - 3)
-19
23