1. Starter
Multiply out the brackets:
(x+3)(x+3)
(x+2)2
(x-5)2
Factorise:
x2 + 8x + 16
x2 - 12x + 36
x2 + 22x + 121
= x2 + 6x + 9
= x2 + 4x + 4
= x2 – 10x + 25
= (x + 4)²
= (x – 6)²
= (x + 11)²

2. x2 + 4x + 4
(x+4)2
2(x+4) A B
(x+2)2
(x+4x)2 C D

3. x2 + 4x + 3
(x+4)2 + 3
(x+2)2 - 1 A B
(x+2)2 + 3
(x+2)2 + 1 C D

4. x2 + 6x + 9
(x+6)2
(x+3)2 A B
(x+9)2
(x+4)2 C D

5. x2 + 6x + 10
(x+6)2 + 1
(x+3)2 + 10 A B
(x+3)2 - 1
(x+3)2 + 1 C D

6. x2 + 10x + 25
(x+5)2 +20
(x+10)2 A B
(x+5)2 +10
(x+5)2 C D

7. x2 + 10x + 24
(x+5)2 +1
(x+5)2 -1 A B
(x+5)2 -2
(x+5)2 +4 C D

8. x2 + 12x + 36
(x+6)2
(x+6)2 + 4 A B
(x+12)2
(x+6)2 +2 C D

9. x2 + 12x + 46
(x+6)2+10
(x+6)2+16 A B
(x+6)2-10
(x+6)2+36 C D

10. x2 + 20x + 80
(x+10)2+20
(x+10)2+10 A B
(x+10)2-10
(x+10)2-20 C D

11. x2 + 14x + 56
(x+7)2+8
(x+8)2+6 A B
(x+7)2+7
(x+7)2-7 C D

12. Half the coefficient of x
Completing the Square
x2 + 10x + 10 = 0
Half the coefficient of x
(x + 5)2 – (5)² + 10 = 0
(x + 5)2 – = 0
Simplify
Minimum point
(-5, -15)
(x + 5)2 – 15 = 0
(x + 5)2 = 15
Solve
x + 5 = ± √15
x = - 5 ± √15
Make sure you have both + and – square root!

13. Half the coefficient of x
Completing the Square
x2 - 8x + 5 = 0
Half the coefficient of x
(x - 4)2 – (-4)² + 5 = 0
(x - 4)2 – = 0
Simplify
Minimum point
(4, -11)
(x - 4)2 – 11 = 0
(x - 4)2 = 11
Solve
x - 4 = ± √11
x = 4 ± √11
Make sure you have both + and – square root!

14. Half the coefficient of x
Completing the Square
x2 - 14x - 9 = 0
Half the coefficient of x
(x - 7)2 – (-7)² - 9 = 0
(x - 7)2 – = 0
Simplify
Minimum point
(7, -58)
(x - 7)2 – 58 = 0
(x - 7)2 = 58
Solve
x - 7 = ± √58
x = 7 ± √58
Make sure you have both + and – square root!

15. Minimum Points (x + 5)2 – 15 = 0 Minimum at (-5, -15)
(x - p)2 + q = 0 Minimum at (p, q)

16. Answers (a) (b) (c) 1 (x + 4)² - 25 x = 1 or -9 (-4, -25) 2
(3, -19) 3
(x + 5)² - 34
x = -5 ± √34
(-5, -34) 4
(x + 3)² - 16
x = 1 or -7
(-3, -16) 5
(x - 5)² - 22
x = 5 ± √22
(5, -22) 6
(x – 7/2)² - 45/4
x = 7/2 ± 3/2√5
(3.5, ) 7
(x + 6)² - 41
x = -6 ± √53
(-6, -41) 8
(x + 3/2)² + 7/4
x = -3/2 ± ½√29
(-1.5, 1.75) 9
4(x + 1)² - 16
x = 1 or -3
(-1, -16) 10
3(x + 1)² - 12
(-1, -12) 11
5(x + 1)² - 21
x = -1 ± √(29/5)
(-1, -21) 12
2(x + 3/2)² + ½
x = -3/2 ± ½√23
(-1.5, 0.5)