1. Expand the following to create 2 quadratic identities:
(x + 3)(x - 3)
(2n + 1)(2n - 1)
(x + 3)(x - 3) ≡ x2 - 9
(2n + 1)(2n - 1) ≡ 4n2 - 1
What do you notice?

2. (x + 3)(x - 3) ≡ x2 - 9 (2n + 1)(2n - 1) ≡ 4n2 - 1
When expanded, the middle terms cancelled out.
This is a special case called the
‘Difference of two squares’ that allows us to expand and factorise certain expressions very quickly.
Can you think why its called the difference of two squares?
x2 – 32
(2n)2 – 12

3. (x + 3)(x - 3) ≡ x2 - 9 (2n + 1)(2n - 1) ≡ 4n2 - 1
When expanded, the middle terms cancelled out.
This is a special case called the
‘Difference of two squares’ that allows us to expand and factorise certain expressions very quickly.
What will these expressions need to have in order for us to apply this case?
We must have a subtraction, for the difference
and we must have square numbers …

4. x2 – 4 x2 + 0x – 4 (x – 2)(x + 2) x2 – 25 x2 + 0x – 25 (x – 5)(x + 5)
Consider …
x2 – 4
x2 + 0x – 4
(x – 2)(x + 2) +2 -2
x2 – 25
x2 + 0x – 25
(x – 5)(x + 5) +5 -5

5. How can we find the area of the coloured shape?
Consider this square… b a
How can we find the area of the coloured shape?
Total Area – White Area

6. How can we find the area of the coloured shape?
Consider this square… b a
How can we find the area of the coloured shape?
a2 – White Area

7. How can we find the area of the coloured shape?
Consider this square… b a
How can we find the area of the coloured shape?
a2 - b2

8. What if we rearrange the shape?
Consider this square… b a
What if we rearrange the shape?
a2 - b2

9. What are the dimensions now?
Consider this square…
Why?
a - b b a a b
What are the dimensions now?
a2 - b2

10. What is the area of the rearranged shape?
Consider this square…
a - b b a a b
What is the area of the rearranged shape?
a2 - b2
(a + b)(a – b)

11. What do you know about the 2 areas?
Consider this square…
a - b b a a b
What do you know about the 2 areas?
a2 - b2
(a + b)(a – b) ≡

12. Factorise the following:1
x2 - 9 2
a2 – 100 3
4h2 - 36 4
16v2 – 49 5
25x2 – 81 6
x2 – 64
Extension:
Factorise x2 - 84

13. Factorise the following:1
x2 - 9
(x + 3)(x - 3) 2
a2 – 100 3
4h2 - 36 4
16v2 – 49 5
25x2 – 81 6
x2 – 64

14. Factorise the following:1
x2 - 9
(x + 3)(x - 3) 2
a2 – 100
(a + 10)(a - 10) 3
4h2 - 36 4
16v2 – 49 5
25x2 – 81 6
x2 – 64

15. Factorise the following:1
x2 - 9
(x + 3)(x - 3) 2
a2 – 100
(a + 10)(a - 10) 3
4h2 - 36
(2h + 6)(2h - 6) 4
16v2 – 49 5
25x2 – 81 6
x2 – 64

16. Factorise the following:1
x2 - 9
(x + 3)(x - 3) 2
a2 – 100
(a + 10)(a - 10) 3
4h2 - 36
(2h + 6)(2h - 6) 4
16v2 – 49
(4v + 7)(4v - 7) 5
25x2 – 81 6
x2 – 64

17. Factorise the following:1
x2 - 9
(x + 3)(x - 3) 2
a2 – 100
(a + 10)(a - 10) 3
4h2 - 36
(2h + 6)(2h - 6) 4
16v2 – 49
(4v + 7)(4v - 7) 5
25x2 – 81
(5x + 9)(5x - 9) 6
x2 – 64

18. Factorise the following:1
x2 - 9
(x + 3)(x - 3) 2
a2 – 100
(a + 10)(a - 10) 3
4h2 - 36
(2h + 6)(2h - 6) 4
16v2 – 49
(4v + 7)(4v - 7) 5
25x2 – 81
(5x + 9)(5x - 9) 6
x2 – 64
(x + 8)(x - 8)

19. Factorise the following:1
x2 - 9
(x + 3)(x - 3) 2
a2 – 100
(a + 10)(a - 10) 3
4h2 - 36
(2h + 6)(2h - 6) 4
16v2 – 49
(4v + 7)(4v - 7) 5
25x2 – 81
(5x + 9)(5x - 9) 6
x2 – 64
(x + 8)(x - 8)
Extension:
(x )(x )

20. Thinking about the difference of two squares can you find the value of the following without using a calculator!
132 – 32
262 – 242
(13 - 3)(13 + 3)
( )( )
10 X 16
2 X 50
160
100