1. 𝑥+5 𝑥+2 𝑥+5 𝑥−2 (𝑥−5)(𝑥+2) (𝑥−5)(𝑥−2)
When each of the following expressions is expanded into the form 𝑥 𝑥 , what will the numbers in and be?
𝑥+5 𝑥+2
𝑥+5 𝑥−2
(𝑥−5)(𝑥+2)
(𝑥−5)(𝑥−2)
What do you notice?

2. 𝑥+6 𝑥+1 𝑥+6 𝑥−1 (𝑥−6)(𝑥+1) (𝑥−6)(𝑥−1)
When each of the following expressions is expanded into the form 𝑥 𝑥 , does what you noticed about the values in and still apply
𝑥+6 𝑥+1
𝑥+6 𝑥−1
(𝑥−6)(𝑥+1)
(𝑥−6)(𝑥−1)

3. 𝑥+𝑎 𝑥+𝑏 Does what you have noticed work in general?
𝑥+𝑎 𝑥+𝑏
Why do I not need all the other combinations as before?
This expands to:
𝑥 2 +𝑎𝑥+𝑏𝑥+𝑎𝑏
Which simplifies to:
𝑥 2 +(𝑎+𝑏)𝑥+𝑎𝑏

4. In general: 𝑥+𝑎 𝑥+𝑏 ≡ 𝑥 2 +(𝑎+𝑏)𝑥+𝑎𝑏
Therefore, when you expand
𝑥+𝑎 𝑥+𝑏
into the form 𝑥 𝑥+
will be the sum of 𝑎 and 𝑏
will be the product of 𝑎 and 𝑏
In general: 𝑥+𝑎 𝑥+𝑏 ≡ 𝑥 2 +(𝑎+𝑏)𝑥+𝑎𝑏
Copy this into your books

5. In your books
Use this identity to expand the brackets below: 𝑥+𝑎 𝑥+𝑏 ≡ 𝑥 2 +(𝑎+𝑏)𝑥+𝑎𝑏

6. (𝑥+𝑎) 2 On your whiteboards Use this identity 𝑥+𝑎 𝑥+𝑏 ≡ 𝑥 2 + 𝑎+𝑏 𝑥+𝑎𝑏
to expand
(𝑥+𝑎) 2
Check you are correct by using a diagram
In general:
(𝑥+𝑎) 2 ≡ 𝑥 2 +2𝑎𝑥+ 𝑎 2
Copy this into your books

7. If you substitute −𝑎 for 𝑎 in the identity, you get:
(𝑥+𝑎) 2 ≡ 𝑥 2 +2𝑎𝑥+ 𝑎 2
If you substitute −𝑎 for 𝑎 in the identity, you get:
(𝑥−𝑎) 2 ≡ 𝑥 2 −2𝑎𝑥+ 𝑎 2
Why is 𝑎 2 still positive?

8. Use these identities to expand the brackets below
In your books
Use these identities to expand the brackets below
(𝑥+𝑎) 2 ≡ 𝑥 2 +2𝑎𝑥+ 𝑎 2
(𝑥−𝑎) 2 ≡ 𝑥 2 −2𝑎𝑥+ 𝑎 2

9. On your whiteboards 𝑥+𝑎 𝑥+𝑏 ≡ 𝑥 2 + 𝑎+𝑏 𝑥+𝑎𝑏
𝑥+𝑎 𝑥+𝑏 ≡ 𝑥 2 + 𝑎+𝑏 𝑥+𝑎𝑏
Substitute −𝑎 for 𝑏 in the identity above.
What new identity do you end up with?
𝑥+𝑎 𝑥−𝑎 ≡ 𝑥 2 − 𝑎 2
Copy this into your books

10. Use the identity to expand the brackets below
In your books
Use the identity to expand the brackets below
𝑥+𝑎 𝑥−𝑎 ≡ 𝑥 2 − 𝑎 2

11. Use these identities to expand the brackets below
In your books
Use these identities to expand the brackets below