1. Two-step with common factor
Factorising
Quadratic
Expressions
Difference of Squares
Trinomials
Complex trinomials
Two-step with common factor

2. Multiply out these brackets (F O I L)
(x – 3)(x + 3)
(b + c)(b - c)
= x2
+ 3x
- 3 x
- 9
= b2
- bc
+ bc
- c2
= b2
- c2
= x2
- 9
= x2
- 32

3. Any expression which can be written as (fred + nelly)(fred – nelly)
Difference of Squares
Any expression which can be written as
a2 – b2
can be factorised
( put into brackets )
a2 – b2 = (a + b)(a – b)
x2 - y2 =
(x + y)(x - y)
p2 - q2 =
(p + q)(p - q)
fred2 – nelly2 =
(fred + nelly)(fred – nelly)

4. Examples
1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100
x2 - 9
k2 - 25
= x2 - 32
= k2 - 52
= (x + 3)(x - 3)
= (k + 5)(k - 5)
a2 - 16b2
4x2 - 1
= a2 -
(4b)2
= (2x)2 - 12
= (a + 4b)(a - 4b)
= (2x + 1)(2x - 1)

5. Exercise 1
1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100
6. 9a2 - 4
1. x2 - 4
(x + 2)(x - 2)
(3a + 2)(3a - 2)
2. t2 - 81
7. 25z2 - 1
(t + 9)(t - 9)
(5z + 1)(5z - 1)
3. b2 - 49
8. 4e2 - 81f2
(b + 7)(b - 7)
(2e + 9f)(2e - 9f)
4. y2 - 25
9. 49g2 - 64h2
(y + 5)(y - 5)
(7g + 8h)(7g - 8h)
5. p2 - 1
q2 - 9r2
(p + 1)(p - 1)
(10q + 3r)(10q - 3r)

6. Two-step Difference of Squares - common factor
Always look for a common factor and take it out before doing anything else
3x2 - 12
= 3(x2 - 4)
= 3(x + 2)(x - 2)
20d2 - 45e2
2a2 - 50b2
= 2(a2 - 25b2)
= 5(4d2 - 9e2)
= 2(a + 5b)(a - 5b)
= 5(2d + 3e)(2d - 3e)

7. Exercise 2
6. 6g2 - 24h2
1. 2w2 - 18
6(g + 2h)(g - 2h)
2(w + 3)(w - 3)
7. 7p2 - 7q2
2. 3y2 - 3
7(p + q)(p - q)
3(y + 1)(y - 1)
8. 27a2 - 12b2
3. 2t2 - 98s2
3(3a + 2b)(3a - 2b)
2(t + 7s)(t - 7s)
9. 32f2 - 2g2
4. 5x2 - 20y2
2(4f + g)(4f - g)
5(x + 2y)(x - 2y)
5. 8m2 - 50n2
k2 - 54t2
2(2m + 5n)(2m - 5n)
6(10k + 3t)(10k - 3t)

8. Factorising is the opposite process to multiplying out two brackets
You probably used FOIL
F  firsts
(x + 3)(x + 2)
O  outers = x2
+ 2x
+ 3x
+ 6
I  inners
= x2 + 5x + 6
L  lasts
So x2 + 5x + 6 factorised is
(x + 3)(x + 2)
3 + 2 = 5
3 × 2 = 6

9. Coefficient of x2 is 1 x2 + 7x + 12 = (x + 3)(x + 4) x2 + 4x - 12
Look for factors of +12 which add to 7
1 and 12
2 and 6
3 and 4
7 =
= (x + 3)(x + 4)
x2 + 4x - 12
1 and 12
2 and 6
3 and 4
Look for factors of -12 which add to 4
4 =
= (x + 6)(x - 2)

10. Coefficient of x2 is 1 x2 - 2x - 35 = (x + 5)(x - 7) x2 + x - 20
Look for factors of - 35 which add to -2
1 and 35
5 and 7
-2 =
= (x + 5)(x - 7)
x2 + x - 20
1 and 20
2 and 10
4 and 5
Look for factors of -20 which add to +1
1 =
= (x + 5)(x - 4)

11. Exercise 3 1. x2 + 5x + 6 6. x2 - 7x + 10 2. x2 + 8x + 15

12. Coefficient of x2 is > 1 Find factors of 12 which add to -7
2x2 - 7x + 6
2 x +6 = 12
Find factors of 12 which add to -7
Split in half and take common factors
= 2x2 - 3x - 4x + 6
-3 and -4: -7x = -3x - 4x
= x(2x - 3) - 2(2x - 3)
coefficients bracket and common bracket
= (x - 2)(2x - 3)

13. Coefficient of x2 is > 1 Find factors of -24 which add to 10
3x2 + 10x - 8
3 x -8 = -24
Find factors of -24 which add to 10
Split in half and take common factors
= 3x2 + 12x - 2x - 8
+12 and -2: 10x = +12x - 2x
= 3x(x + 4) - 2(x + 4)
coefficients bracket and common bracket
= (3x - 2)(x + 4)

14. Coefficient of x2 is > 1 Find factors of -20 which add to -8
4x2 - 8x - 5
4 x -5 = -20
Find factors of -20 which add to -8
Split in half and take common factors
= 4x2 - 10x + 2x - 5
-10 and +2: 8x = -10x + 2x
= 2x(2x - 5) + 1(2x - 5)
coefficients bracket and common bracket
= (2x + 1)(2x - 5)

15. Coefficient of x2 is > 1 Find factors of -30 which add to +1
2x2 + x - 15
2 x -15 = -30
Find factors of -30 which add to +1
Split in half and take common factors
= 2x2 + 6x - 5x - 15
+6 and -5: x = +6x - 5x
= 2x(x + 3) - 5(x + 3)
coefficients bracket and common bracket
= (2x - 5)(x + 3)

16. Exercise 4 1. 2x2 + 11x + 12 6. 3x2 + 2x - 8 2. 3x2 + 13x + 12