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

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

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

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

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

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

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

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

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

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

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

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 – 25 + 10 = 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!

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 – 16 + 5 = 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!

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 – 49 - 9 = 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!

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

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, -11.25) 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)