1. World 1-9
Completing the Square

2. 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

3. 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

4. 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)

5. 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

6. Homework
World 1-9 Worksheet