Difference of Two Squares Demonstration

1. Difference of Two Squares – Demonstration
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2. Difference of Two Squares
𝑥
𝑥
Area = 𝑥2 − 𝑦2
𝑥 − 𝑦
𝑥 + 𝑦
𝑦
𝑦
A ‘perfect’ square: 𝑥2
We subtract another square: 𝑦2
What is the area of the new shape?
What is the difference of two squares?
Area = (𝑥 + 𝑦)(𝑥 − 𝑦)
This rearrangement shows how
we can factorise a square that
has had another square
subtracted from it.
𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦)

3. Difference of Two Squares
𝑥
𝑥
Area = 𝑥2 − 49
𝑥 − 7
𝑥 + 7
7
7
A ‘perfect’ square: 𝑥2
We subtract another square: 72
What is the difference of two squares?
Area = (𝑥 + 7)(𝑥 − 7)
This rearrangement shows how
we can factorise a square that
has had another square
subtracted from it.
𝑥2 − 72 = (𝑥 + 7)(𝑥 − 7)
= 𝑥2 − 72

4. Difference of Two Squares
𝑥
𝑥
Area = 𝑥2 − 49
𝑥 − 7
𝑥 + 7
7
7
Area = (𝑥 + 7)(𝑥 − 7)
= 𝑥2 − 72
(𝑥 + 7)(𝑥 − 7)
If we expand:
𝑥2 + 7𝑥 − 7𝑥 − 49
𝑥2 − 49

5. EXAMPLE
Factorise each expression by sketching the difference of two squares.
𝑎2 − 𝑏2 =
𝑥2 − 4𝑦2 =
9𝑥2 − 16𝑦2 =
𝑥2 − 9 =
𝑥2 − 22 =
𝑥2 − 25 =

6. (𝑥 + 3)(𝑥 − 3) 𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
(𝑥 + 2)(𝑥 − 2) 𝑥2 − 4𝑦2 = (𝑥 + 2𝑦)(𝑥 − 2𝑦)
(𝑥 + 5)(𝑥 − 5) 9𝑥2 − 16𝑦2 = (3𝑥 + 4𝑦)(3𝑥 − 4𝑦)
EXAMPLE
Factorise each expression by sketching the difference of two squares.
𝑥2 − 9 =
𝑥2 − 22 =
𝑥2 − 25 =

7. We need to identify one square subtracted from another.
𝑥2 − 42
Difference of Two Squares
DEMO
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 4
( )
+ −
𝑥 4 4
𝑥
(𝑥 + 4)(𝑥 − 4)
(4)2

8. We need to identify one square subtracted from another.
𝑥2 − 72
Difference of Two Squares
DEMO
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 7
( )
+ −
𝑥 7 7
𝑥
(𝑥 + 7)(𝑥 − 7)
(7)2

9. We need to identify one square subtracted from another.
𝑥2 − 36
Difference of Two Squares
DEMO
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 6
( )
+ −
𝑥 6 6
𝑥
(𝑥 + 6)(𝑥 − 6)
(6)2

10. We need to identify one square subtracted from another.
𝑥2 − 36
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 6
( )
+ −
𝑥 6 6
𝑥
(𝑥 + 6)(𝑥 − 6)
(6)2
𝑥2 − 4
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 2
( )
+ −
𝑥 2 2
𝑥
(𝑥 + 2)(𝑥 − 2)
(2)2

11. We need to identify one square subtracted from another.
𝑥2 − 36
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 6
( )
+ −
𝑥 6 6
𝑥
(𝑥 + 6)(𝑥 − 6)
(6)2
𝑥2 − 25
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 5
( )
+ −
𝑥 5 5
𝑥
(𝑥 + 5)(𝑥 − 5)
(5)2

12. We need to identify one square subtracted from another.
𝑥2 − 36
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 6
( )
+ −
𝑥 6 6
𝑥
(𝑥 + 6)(𝑥 − 6)
(6)2
9 − 𝑦2
Factorise:
3 2 −
𝒂 = 3
𝒃 = 𝑦
( )
+ −
3 𝑦 𝑦
3
(3 + 𝑦)(3 − 𝑦)
𝑦 2

13. We need to identify one square subtracted from another.
𝑥2 − 36
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 6
( )
+ −
𝑥 6 6
𝑥
(𝑥 + 6)(𝑥 − 6)
(6)2
𝑥2 − 4𝑦2
Factorise:
𝑥 2 −
𝒂 = 𝑥
𝒃 = 2𝑦
( )
+ −
𝑥 2𝑦 2𝑦
𝑥
(𝑥 + 2𝑦)(𝑥 − 2𝑦)
2𝑦 2

14. We need to identify one square subtracted from another.
(𝑥 + 3)2 − 𝑥2
Difference of Two Squares
DEMO
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝒂 = 𝑥 + 3
𝒃 = 𝑥
( )
+ −
𝑥
𝑥 + 3 𝑥 + 3 𝑥
(2𝑥 + 3)(3)
6𝑥 + 9
3(2𝑥 + 3)

15. We need to identify one square subtracted from another.
Difference of Two Squares
DEMO
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
(𝑥 + 3)2 − (𝑥 + 2)2
Factorise:
𝒂 = 𝑥 + 3
𝒃 = 𝑥 + 2
(2𝑥 + 5)(1)
2𝑥 + 5
(𝑥 + 3 + 𝑥 + 2)(𝑥 + 3 − 𝑥 − 2)

16. 𝒂 = 𝑥
We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝒃 = 𝑥 − 1
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
Why a positive?
(𝑥 − (𝑥 − 1))

17. We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
(𝑥 + 2)2 − 𝑥2
Factorise:
𝒂 = 𝑥 + 2
𝒃 = 𝑥
( )
+ −
𝑥
𝑥 + 2 𝑥 + 2 𝑥
(2𝑥 + 2)(2)
4𝑥 + 4
4(𝑥 + 1)
𝒂 = 𝑥
𝒃 = 𝑥 − 1

18. We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
𝒂 = 𝑥
𝒃 = 𝑥 − 1
(𝑥 + 3)2 − 𝑥2
Factorise:
𝒂 = 𝑥 + 3
𝒃 = 𝑥
( )
+ −
𝑥
𝑥 + 3 𝑥 + 3 𝑥
(2𝑥 + 3)(3)
6𝑥 + 9
3(2𝑥 + 3)

19. We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
𝒂 = 𝑥
𝒃 = 𝑥 − 1
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
𝑥2 − (𝑥 − 3)2
Factorise:
𝒂 = 𝑥
𝒃 = 𝑥 − 3
( )
+ −
𝑥 𝑥 − 3 𝑥 − 3
𝑥
(2𝑥 − 3)(3)
6𝑥 − 9
3(2𝑥 − 3)

20. We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
𝑥2 − (𝑥 − 5)2
Factorise:
𝒂 = 𝑥
𝒃 = 𝑥 − 5
( )
+ −
𝑥 𝑥 − 5 𝑥 − 5
𝑥
(2𝑥 − 5)(5)
10𝑥 − 25
5(2𝑥 − 5)
𝒂 = 𝑥
𝒃 = 𝑥 − 1

21. We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
(𝑥 + 1)2 − (𝑥 − 2)2
Factorise:
𝒂 = 𝑥 + 1
𝒃 = 𝑥 − 2
(2𝑥 − 1)(3)
6𝑥 − 3
(𝑥 + 1 + 𝑥 − 2)(𝑥 + 1 − 𝑥 + 2)
𝒂 = 𝑥
𝒃 = 𝑥 − 1
3(2𝑥 − 1)

22. We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
Factorise these expressions into two brackets.
Factorise & simplify these expressions.
𝑥2 − 𝑦2 =
𝑥2 − 32 =
49 − 𝑏2 =
16𝑦2 − 81 =
𝑥 + 4 2 − 𝑥2 =
𝑥 + 3 2
− 𝑥 + 2 2
=
𝑥 + 2 2
− 𝑥 − 5 2
=
𝑥2 − 𝑥 − 4 2 =
𝒂 = 𝑥
𝒃 = 𝑥 − 1

23. We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
Factorise these expressions into two brackets.
Factorise & simplify these expressions.
𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦)
𝑥2 − 32 = (𝑥 + 3)(𝑥 − 3)
49 − 𝑏2 = (7 + 𝑏)(7 − 𝑏)
16𝑦2 − 81 = (4𝑦 + 9)(4𝑦 − 9)
𝑥 + 4 2 − 𝑥2 =
𝑥 + 3 2
− 𝑥 + 2 2
=
𝑥 + 2 2
− 𝑥 − 5 2
=
𝑥2 − 𝑥 − 4 2 =
𝒂 = 𝑥
𝒃 = 𝑥 − 1

24. We need to identify one square subtracted from another.
𝑥2 − (𝑥 − 1)2
Difference of Two Squares
DEMO YOUR TURN
𝒂2 − 𝒃2 = (𝒂 + 𝒃)(𝒂 − 𝒃)
Factorise:
( )
+ −
𝑥 𝑥 − 1 𝑥 − 1
𝑥
(2𝑥 − 1)(1)
2𝑥 − 1
Factorise these expressions into two brackets.
Factorise & simplify these expressions.
𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦)
𝑥2 − 32 = (𝑥 + 3)(𝑥 − 3)
49 − 𝑏2 = (7 + 𝑏)(7 − 𝑏)
16𝑦2 − 81 = (4𝑦 + 9)(4𝑦 − 9)
𝑥 + 4 2 − 𝑥2 = 8(𝑥 + 2)
𝑥 + 3 2
− 𝑥 + 2 2
= 2𝑥 + 5
𝑥 + 2 2
− 𝑥 − 5 2
= 7(2𝑥 − 3)
𝑥2 − 𝑥 − 4 2 = 8(𝑥 − 2)
𝒂 = 𝑥
𝒃 = 𝑥 − 1

25. Questions?
Comments?
Suggestions?
…or have you found a mistake!?
Any feedback would be appreciated .
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tom@goteachmaths.co.uk