Expanding Brackets special Perfect Squares A perfect square

Expanding Brackets - special

Perfect Squares A perfect square in brackets form looks like (x – 7) or more simply as (x – 7)2 It is known as a perfect square as the expression in each bracket is the same! What pattern do you get when you expand a perfect square? Expand the following two expressions, and check. • (x + 3)2 • (x – 3)2

Perfect Squares A perfect square can be shown algebraically as (x + a)2 Expand out (x + a)2 = (x + a) = x 2 + ax + a 2 = x 2 + 2 ax + a 2 What do you think you would get for (x - a)2? Generalised form is (x - a)2 = x 2 - 2 ax + a 2 (x ± a)2 = x 2 ± 2 ax + a 2

Difference of Two Squares If you took a large square, and subtracted a smaller square from one corner, what is the number expression for the remaining area? 6 2 6 - 2 You could write the answer as 62 – 22 = (the difference of 2 squares)

Difference of Two Squares 6 6 2 2 2 ② Rotate and move here ① Cut here + 2 Therefore our expression for the remaining shape becomes (6 + 2)(6 – 2) So (6 + 2)(6 – 2) = 62 – 22

Difference of Two Squares Generalising algebraically: x x a a a ② Rotate and move here ① Cut here + a Therefore our expression for the remaining shape becomes (x + a)(x – a) So (x + a)(x – a) = x 2 – a 2

Summary For perfect squares (x ± a)2 = x 2 ± 2 ax + a 2 For difference in two squares (x + a)(x – a) = x 2 – a 2

Practice Expand simplify if required: 1. (x + 4) 2. (x - 5)2 3. (3 x + 2) 4. (5 x – 3)2 5. 2(x + 3)2 6. (x – 4)(x + 4) 7. (x + 7)(x - 7) 8. (2 x + 1)(2 x – 1) 9. (4 – x)(x + 4) 10. (x + 15)(x - 15) 11. (x – ⅓)(x + ⅓) 12. (3 x + 2 k)(3 x – 2 k)