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 2 + 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?

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 2 + 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)

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

In general: 𝑥+𝑎 𝑥+𝑏 ≡ 𝑥 2 +(𝑎+𝑏)𝑥+𝑎𝑏 Therefore, when you expand 𝑥+𝑎 𝑥+𝑏 into the form 𝑥 2 + 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

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

(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

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?

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

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

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

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