World 1-9 Completing the Square

Factor these perfect square trinomials Recall that the numbers 4, 9,16, 25 etc. are known as perfect squares. It is also possible to have what is called a perfect square trinomial. Geometrically it is a square. eg. 1 Factor these perfect square trinomials x2 + 6x + 9 x2 - 8x + 16 25x2 + 40xy + 16y2 =(x + )(x + ) 3 3 =(x - )(x - ) 4 4 =(5x + )(5x + ) 4y 4y =(x + 3)2 =(x - 4)2 =(5x +4y)2 x + 3 x - 4 5x + 4y

If the trinomial is in the form ax2 + bx + c it can be modified into the form a(x –h)2 + k. This called completing the square! Case 1: a ≠ 1 & answer is a difference of squares = ax2 + bx + c Step by Step = 2x2 + 16x - 40 = 2(x2 + 8x - 20) 1. Factor out the a = 2(x2 + 8x +16 – 16 - 20) 2. Take half of this b and square it Add this # and subtract this # = 2((x + )2 4 - 36) 3. Factor the perfect square trinomial = 2( ((x+4) -6) ((x+4) +6) 4. Factor if a difference of squares = 2 (x-2) (x+10) 5. Simplify

Case 2: a =1 and answer is a difference of squares  = x2 + bx + c = x2 + 12x - 13 = x2 + 12x +36 – 36 - 13 = (x +6)2 - 49 = ((x+6) -7) ((x+6) +7) = (x-1) (x+13)

Case 3: a ≠ 1 and dealing with fractions ax2 + bx + c = 4x2 + 5x - 9 5 4 9 4 = 4(x2 + x - ) 5 4 25 64 25 64 9 4 = 4(x2 + x + - - ) 5 8 = 4((x + )2 25 64 - - ) 144 64 5 8 = 4((x + )2 169 64 - ) 5 8 13 8 5 8 13 8 = 4( ((x+ ) - ) ((x+ )+ ) 8 18 8 = 4( (x - ) (x + ) 9 4 = 4 (x - ) (x + ) 1

Homework World 1-9 Worksheet