Factorising Quadratics

Factorising Quadratics by inspection

1. Factorising Quadratics
Index
1. What are quadratics?
2. Factorising quadratics (coefficient of x
2 is 1)
3. Predicting the signs of the final answer.
4. Factorising Quadratics (coefficient of x
2 is not 1)

2. Factorising Quadratics
What are they?
Remember expanding two bracket problems

3. Factorising Quadratics
What are they?
Remember expanding two bracket problems
(x + 3)(x + 4)
= x×x

4. Factorising Quadratics
What are they?
Remember expanding two bracket problems
(x + 3)(x + 4)
= x×x + x×4

5. Factorising Quadratics
What are they?
Remember expanding two bracket problems
(x + 3)(x + 4)
= x×x + 3×x + x×4

6. Factorising Quadratics
What are they?
Remember expanding two bracket problems
(x + 3)(x + 4)
= x×x + 3×x + x×4 + 3×4

7. Factorising Quadratics
What are they?
Remember expanding two bracket problems
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x + x×4 + 3×4

8. Factorising Quadratics
What are they?
Remember expanding two bracket problems
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x
add the like terms
= x
2


+

7x + 12
+ x×4 + 3×4

9. Factorising Quadratics
What are they?
Remember expanding two bracket problems
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x
add the like terms
= x
2


+

7x + 12
+ x×4 + 3×4
In this example the resulting equation is called a QUADRATIC EQUATION.
The biggest power of x is 2 in Quadratic Equation.
Examples of quadratic equations:
x
2
+

x + 2 x
2
–

2x + 6 4x
2
– 100 2x
2
–

14x + 20

10. Factorising Quadratics
Factorising Quadratic Expressions
Factorising is reversing the process of removing brackets.
To factorise a quadratic you need to put back the two brackets.
Let’s take a closer look at how the quadratic was formed.
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x
= x
2


+

7x + 12
+ x×4 + 3×4

11. Factorising Quadratics
Factorising Quadratic Expressions
Factorising is reversing the process of removing brackets.
To factorise a quadratic you need to put back the two brackets.
Let’s take a closer look at how the quadratic was formed.
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x
= x
2


+

7x + 12
+ x×4 + 3×4
Where does the 3
rd
term come from?

12. Factorising Quadratics
Factorising Quadratic Expressions
Factorising is reversing the process of removing brackets.
To factorise a quadratic you need to put back the two brackets.
Let’s take a closer look at how the quadratic was formed.
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x
= x
2


+

7x + 12
+ x×4 + 3×4
Where does the 3
rd
term come from?
multiply the last terms of each bracket

13. Factorising Quadratics
Factorising Quadratic Expressions
Factorising is reversing the process of removing brackets.
To factorise a quadratic you need to put back the two brackets.
Let’s take a closer look at how the quadratic was formed.
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x
= x
2


+

7x + 12
+ x×4 + 3×4
Where does the 3
rd
term come from?
multiply the last terms of each bracket
Where does the middle term come from?

14. Factorising Quadratics
Factorising Quadratic Expressions
Factorising is reversing the process of removing brackets.
To factorise a quadratic you need to put back the two brackets.
Let’s take a closer look at how the quadratic was formed.
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x
= x
2


+

7x + 12
+ x×4 + 3×4
Where does the 3
rd
term come from?
multiply the last terms of each bracket
Where does the middle term come from?
add the last two terms of each bracket

15. Factorising Quadratics
Factorising Quadratic Expressions
Factorising is reversing the process of removing brackets.
To factorise a quadratic you need to put back the two brackets.
Let’s take a closer look at how the quadratic was formed.
(x + 3)(x + 4)
= x×x
= x
2
+ 4x + 3x + 12
+ 3×x
= x
2


+

7x + 12
+ x×4 + 3×4
Where does the 3
rd
term come from?
multiply the last terms of each bracket
Where does the middle term come from?
add the last two terms of each bracket

16. Factorising Quadratics
Factorising Quadratic Expressions
Now let’s start with a quadratic equation and try to find the two brackets
x
2


+

5x + 6

17. Factorising Quadratics
Factorising Quadratic Expressions
Now let’s start with a quadratic equation and try to find the two brackets
= (x )(x )
x
2


+

5x + 6
+ +
Your answer will always look like this

18. Factorising Quadratics
Factorising Quadratic Expressions
Now let’s start with a quadratic equation and try to find the two brackets
= (x )(x )
x
2


+

5x + 6
Your task is to find two numbers so that
when you multiply them you get the last term
and
when you add them you get the middle term
+ +
Your answer will always look like this

19. Factorising Quadratics
Factorising Quadratic Expressions
Now let’s start with a quadratic equation and try to find the two brackets
= (x )(x )
x
2


+

5x + 6
Your task is to find two numbers so that
when you multiply them you get the last term
and
when you add them you get the middle term
+ +
Your answer will always look like this
For this example you must find two numbers that
multiplied together give 6 (write down the factors of 6)
and
added together gives 5 (circle the two numbers)
write these two numbers in the brackets
factors of 6
1 6
2 3

20. Factorising Quadratics
Factorising Quadratic Expressions
Now let’s start with a quadratic equation and try to find the two brackets
= (x )(x )
x
2


+

5x + 6
Your task is to find two numbers so that
when you multiply them you get the last term
and
when you add them you get the middle term
+ 2+ 3
Your answer will always look like this
For this example you must find two numbers that
multiplied together give 6 (write down the factors of 6)
and
added together gives 5 (circle the two numbers)
write these two numbers in the brackets
factors of 6
1 6
2 3

21. Factorising Quadratics
Factorising Quadratic Expressioins
Example 2: Factorise the following quadratic equation
= (x )(x )
x
2


+

9x + 8
+ +
Your answer will always look like this

22. Factorising Quadratics
Factorising Quadratic Expressioins
Example 2: Factorise the following quadratic equation
= (x )(x )
x
2


+

9x + 8
Your task is to find two numbers so that
their product is the last term
and
their sum is the middle term
+ +
Your answer will always look like this

23. Factorising Quadratics
Factorising Quadratic Expressioins
Example 2: Factorise the following quadratic equation
= (x )(x )
x
2


+

9x + 8
Your task is to find two numbers so that
their product is the last term
and
their sum is the middle term
+ +
Your answer will always look like this
For this example you must find two numbers that
multiplied together give 8 (write down the factors of 8)
and
added together gives 9 (circle the two numbers)
write these two numbers in the brackets
factors of 8
1 8
2 4

24. Factorising Quadratics
Factorising Quadratic Expressioins
Example 2: Factorise the following quadratic equation
= (x )(x )
x
2


+

9x + 8
Your task is to find two numbers so that
their product is the last term
and
their sum is the middle term
+ 1+ 8
Your answer will always look like this
For this example you must find two numbers that
multiplied together give 8 (write down the factors of 8)
and
added together gives 9 (circle the two numbers)
write these two numbers in the brackets
factors of 8
1 8
2 4

25. Factorising Quadratics
Factorising Quadratic Expressions
Example 3: Factorise the following quadratic equation
= (x )(x )
x
2


+

9x + 18
+ +
Your answer will always look like this

26. Factorising Quadratics
Factorising Quadratic Expressions
Example 3: Factorise the following quadratic equation
= (x )(x )
x
2


+

9x + 18
Your task is to find two numbers so that
their product is the last term
and
their sum is the middle term
+ +
Your answer will always look like this

27. Factorising Quadratics
Factorising Quadratic Expressions
Example 3: Factorise the following quadratic equation
= (x )(x )
x
2


+

9x + 18
Your task is to find two numbers so that
their product is the last term
and
their sum is the middle term
+ +
Your answer will always look like this
For this example you must find two numbers that
multiplied together give 18 (write down the factors of 18)
and
added together gives 9 (circle the two numbers)
write these two numbers in the brackets
factors of 18
1 18
2 9
3 6

28. Factorising Quadratics
Factorising Quadratic Expressions
Example 3: Factorise the following quadratic equation
= (x )(x )
x
2


+

9x + 18
Your task is to find two numbers so that
their product is the last term
and
their sum is the middle term
+ 3+ 6
Your answer will always look like this
For this example you must find two numbers that
multiplied together give 18 (write down the factors of 18)
and
added together gives 9 (circle the two numbers)
write these two numbers in the brackets
factors of 18
1 18
2 9
3 6

29. Both +
Factorising Quadratics
Predicting the signs
What happens when there are negative numbers in the equation?
Here are the various options: First look at the 2
nd
sign, then the 1
st
sign.
x
2


+

5x + 6
If the 2
nd
sign is + the both signs of the
brackets will be the SAME
The 1
st
sign tells you
that both signs will be +.
= (x

+

2)(x + 3)

30. Both –
Factorising Quadratics
Predicting the signs
What happens when there are negative numbers in the equation?
Here are the various options: First look at the 2
nd
sign, then the 1
st
sign.
x
2
– 5x + 6
If the 2
nd
sign is + the both signs of the
brackets will be the SAME
The 1
st
sign tells you
that both signs will be – .
= (x –

2)(x – 3)

31. Larger
number
is –
Factorising Quadratics
Predicting the signs
What happens when there are negative numbers in the equation?
Here are the various options: First look at the 2
nd
sign, then the 1
st
sign.
x
2
– x – 6
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be – .
= (x +

2)(x – 3)

32. Larger
number
is +
Factorising Quadratics
Predicting the signs
What happends when there are negative numbers in the equation?
Here are the various options: First look at the 2
nd
sign, then the 1
st
sign.
x
2
+ x – 6
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be + .
= (x –

2)(x + 3)

33. Factorising Quadratics
Predicting the signs
What happends when there are negative numbers in the equation?
Here are the various options: First look at the 2
nd
sign, then the 1
st
sign.
x
2
+ x + 6Both will be + (x + )(x + )
x
2
– x + 6Both will be – (x – )(x – )
x
2
– x – 6
Larger number will be –
Smaller number will be +
(x + )(x – )
x
2
+ x – 6
Larger number will be +
Smaller number will be –
(x – )(x + )

34. Factorising Quadratics
Predicting the signs
Example 4: Factorise the following quadratic equation
= (x )(x ) x
2
–

3x – 10 + –
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be – .

35. Factorising Quadratics
Predicting the signs
Example 4: Factorise the following quadratic equation
= (x )(x )
x
2
–

3x – 10
+ 2– 5
For this example you must find two numbers that
multiplied together give –10 (write down the factors of 10)
and
added together gives –3 : the larger number will be negative,
the smaller will be positive (circle the two numbers)
write these two numbers in the brackets
factors of 10
1 –10
2 –5
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be – .
= (x )(x ) + –

36. Factorising Quadratics
Predicting the signs
Example 4: Factorise the following quadratic equation
= (x )(x )
x
2
–

3x – 10
+ 2– 5
For this example you must find two numbers that
multiplied together give –10 (write down the factors of 10)
and
added together gives –3 : the larger number will be negative,
the smaller will be positive (circle the two numbers)
write these two numbers in the brackets
factors of 10
1 –10
2 –5
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be – .
= (x )(x ) + 2– 5

37. Factorising Quadratics
Predicting the signs
Example 4: Factorise the following quadratic equation
= (x )(x ) x
2
–

7x + 6 – –
If the 2
nd
sign is + the signs will be the SAME
The 1
st
sign tells you
that both will be –.

38. Factorising Quadratics
Predicting the signs
Example 4: Factorise the following quadratic equation
= (x )(x ) x
2
–

7x + 6 – –
For this example you must find two numbers that
multiplied together give 6 (write down the factors of 6)
and
added together gives –7 : both numbers are negative
(circle the two numbers)
write these two numbers in the brackets
factors of 6
–1 –6
–2 –3
If the 2
nd
sign is + the signs will be the SAME
The 1
st
sign tells you
that both will be –.

39. Factorising Quadratics
Predicting the signs
Example 4: Factorise the following quadratic equation
= (x )(x ) x
2
–

7x + 6 – 1 – 6
For this example you must find two numbers that
multiplied together give 6 (write down the factors of 6)
and
added together gives –7 : both numbers are negative
(circle the two numbers)
write these two numbers in the brackets
factors of 6
–1 –6
–2 –3
If the 2
nd
sign is + the signs will be the SAME
The 1
st
sign tells you
that both will be –.

40. Factorising Quadratics
Predicting the signs
Example 6: Factorise the following quadratic equation
= (x )(x ) x
2


+

x – 12 – +
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be + .

41. Factorising Quadratics
Predicting the signs
Example 6: Factorise the following quadratic equation
= (x )(x ) x
2


+

x – 12 – +
For this example you must find two numbers that
multiplied together give –12 (write down the factors of 12)
and
added together gives –1 : the larger number will be positive,
the smaller will be negative (circle the two numbers)
write these two numbers in the brackets
factors of 12
–1 12
–2 6
–3 4
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be + .

42. Factorising Quadratics
Predicting the signs
Example 6: Factorise the following quadratic equation
= (x )(x ) x
2


+

x – 12 – 3+ 4
For this example you must find two numbers that
multiplied together give –12 (write down the factors of 12)
and
added together gives +1 : the larger number will be positive,
the smaller will be negative (circle the two numbers)
write these two numbers in the brackets
factors of 12
–1 12
–2 6
–3 4
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be + .

43. Factorising Quadratics
Predicting the signs
Example 7: Factorise the following quadratic equation
= (x )(x ) x
2


+

2x – 8 – +
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be + .

44. Factorising Quadratics
Predicting the signs
Example 7: Factorise the following quadratic equation
= (x )(x ) x
2


+

2x – 8 – +
For this example you must find two numbers that
multiplied together give –8 (write down the factors of 8)
and
added together gives +2 : the larger number will be
positive, the smaller will be negative (circle the two
numbers)
write these two numbers in the brackets
factors of –8
–1 8
–2 4
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be + .

45. Factorising Quadratics
Predicting the signs
Example 7: Factorise the following quadratic equation
= (x )(x ) x
2


+

2x – 8 – 2+ 4
For this example you must find two numbers that
multiplied together give –8 (write down the factors of 8)
and
added together gives +2 : the larger number will be
positive, the smaller will be negative (circle the two
numbers)
write these two numbers in the brackets
factors of –8
–1 8
–2 4
If the 2
nd
sign is – the signs will be OPPOSITE
The 1
st
sign tells you
that the larger factor will
be + .

46. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
Method:
Using the quadratic can be written as ax
2
+ bx +c
1. Look for two numbers that:
• multiply to ac and
• add to b
Call these numbers p and q
2. Write ax
2
+ bx +c as ax
2
+ px + qx +c
3. Now factorise ax
2
+ px + qx +c in two stages

47. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
12x
2


+

x – 6
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
E.g.1

48. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
12x
2


+

x – 6
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
ac = 12 × –6 = –72
ac = –72 b = 1
E.g.1

49. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
12x
2


+

x – 6
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of -72
–1 72
–2 36
–3 14
–4 18
–6 12
–8 9
ac = 12 × –6 = –72
ac = –72 b = 1
–8 × 9 = – 72
– 8 + 9 = 1
p = –8
q = 9
E.g.1

50. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
12x
2


+

x – 6
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of -72
–1 72
–2 36
–3 14
–4 18
–6 12
–8 9
ac = 12 × –6 = –72
ac = –72 b = 1
–8 × 9 = – 72
– 8 + 9 = 1
p = –8
q = 9 = 12x
2
– 8x +

9x – 6
Now factorise ax
2
+ px + qx +c
in two stages
12x
2


+

x – 6
E.g.1

51. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
12x
2


+

x – 6
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of -72
–1 72
–2 36
–3 14
–4 18
–6 12
–8 9
ac = 12 × –6 = –72
ac = –72 b = 1
–8 × 9 = – 72
– 8 + 9 = 1
p = –8
q = 9 = 12x
2
– 8x +

9x – 6
Now factorise ax
2
+ px + qx +c
in two stages
= 4x(3x – 2) +

3(3x – 2)
= (3x – 2)(4x +

3)
12x
2


+

x – 6
E.g.1

52. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
3x
2
+ 7 x + 2
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
E.g.2

53. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
3x
2
+ 7 x + 2
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
ac = 3 × 2 = 6
ac = 6 b = 7
E.g.2

54. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
3x
2
+ 7 x + 2
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of 6
1 6
2 3
ac = 3 × 2 = 6
ac = 6 b = 7
1 × 6 = 6
1 + 6 = 7
p = 1
q = 6
E.g.2

55. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
3x
2
+ 7 x + 2
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of 6
1 6
2 3
ac = 3 × 2 = 6
ac = 6 b = 7
1 × 6 = 6
1 + 6 = 7
p = 1
q = 6 = 3x
2
+ 1x +

6x + 2
Now factorise ax
2
+ px + qx +c
in two stages
3x
2


+ 7

x + 2
E.g.2

56. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
3x
2
+ 7 x + 2
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of 6
1 6
2 3
ac = 3 × 2 = 6
ac = 6 b = 7
1 × 6 = 6
1 + 6 = 7
p = 1
q = 6 = 3x
2
+ 1x +

6x + 2
Now factorise ax
2
+ px + qx +c
in two stages
= x(3x + 1) +

2(3x + 1)
= (3x + 1)(x +

2)
3x
2


+ 7

x + 2
E.g.2

57. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
10x
2
– 13 x – 3
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
E.g.3

58. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
10x
2
– 13 x – 3
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
ac = 10 × –3 = –30
ac = –30 b = –13
E.g.3

59. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
10x
2
– 13 x – 3
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of 30
1 –30
2 –15
3 –10
5 –6
ac = 10 × –3 = –30
ac = –30 b = –13
2 × –15 = –30
2 + –15 = –13
p = 2
q = –15
E.g.3

60. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
10x
2
– 13 x – 3
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of 30
1 –30
2 –15
3 –10
5 –6
ac = 10 × –3 = –30
ac = –30 b = –13
2 × –15 = –30
2 + –15 = –13
p = 2
q = –15 = 10x
2
+ 2x – 15x – 3
Now factorise ax
2
+ px + qx +c
in two stages
10x
2


– 13

x – 3
E.g.3

61. Factorising Quadratics
Factorising Quadratic Expressions
where the coefficient of x
2
is not 1
10x
2
– 13 x – 3
For the equation ax
2
+ bx +c
Your task is to find two numbers (call
them p and q) so that
when you multiply them you get the ac
and when you add them you get the b
factors of 30
1 –30
2 –15
3 –10
5 –6
ac = 10 × –3 = –30
ac = –30 b = –13
2 × –15 = –30
2 + –15 = –13
p = 2
q = –15 = 10x
2
+ 2x – 15x – 3
Now factorise ax
2
+ px + qx +c
in two stages
= 2x(5x + 1) – 3(5x + 1)
= (5x + 1)(2x – 3 )
10x
2


– 13

x – 3
E.g.3