1. Factoring Special Cases
Factor completely:
9x2 + 66x + 21

2. Difference of Perfect Squares
For all numbers a and b, a2 – b2 = (a-b) (a+b) Example 1: (x2 - 25) Since 25 is a perfect square: a2 = x2 and b2 = 25 Therefore: a = ___ and b = ___ Solution: (x2 - 25) = ( ) ( ) x
x x + 5

3. Example 2: 10x2 - 40 10 (x2 – 4) First factor out the 10: __________
Now we have a difference of perfect squares ( )
Note that it has to be a difference so the inner and outer products can cancel each other out.
Solution: ( ) ( )
x2 and 4
10 x x+2

4. Perfect Square Trinomials
Example 3: x2 + 12x + 36 (ax2 + bx + c)
To identify a perfect square trinomial,
See if a (1) and c (36) are perfect squares.
Multiply square roots of a & c & 2. ________________
If the answer equals b (12), you have a perfect square trinomial. __________________________________
Factor by adding/subtracting the square roots.
Solution: x2 +12x +36 = ( ) ( )
= ( )
1 x 6 x 2 = 12
x x + 6 x

5. Example 4: 4x2 – 24x + 36 Yes 2x – 6 2x - 6 2x – 6 2
Are a( ) and c( ) perfect squares? ____________
Does √a x √c x 2 = b? ___________________________
Do we have a perfect square trinomial? ____________
Do we add or subtract the square roots? ____________
Solution: 4x2 – 24x + 36 = ( ) ( )
= ( )
Yes
Yes: 2 x 6 x 2 = 24
Yes
Subtract
2x – x - 6
2x – 6 2

6. Perfect Square Trinomials
What are the factors?
x2 + 6x + 9 =
x2 - 10x + 25 =
x2 + 12x + 36 =
(x+3)2
(x-5)2
(x+6)2

7. Solving Quadratic Equations by Completing the Square
Is x2 - 10x + 25 a Perfect Square Trinomial?
How do you know?
What are its Factors?

8. Creating a Perfect Square Trinomial
In the following trinomial, the constant term is missing X2 + 14x + ____
Find the constant term by squaring half the coefficient of the linear term to make it a perfect square trinomial.
(14/2) X2 + 14x + 49

9. Perfect Square Trinomials
Create perfect square trinomials.
x2 + 20x + ___
x2 - 4x + ___
x2 + 5x + ___
100 4
25/4

10. Solving Quadratic Equations by Completing the Square
Solve the following equation by completing the square:
Step 1: Move quadratic term, and linear term to left side of the equation and the constant to the right side.

11. Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.

12. Solving Quadratic Equations by Completing the Square
Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
(x+4) (x+4) = 36
(x+4)2 = 36

13. Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side

14. Solving Quadratic Equations by Completing the Square
Step 5: Set up the two possibilities and solve

15. Completing the Square-Example #2
Solve the following equation by completing the square:
Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.
2n2 + 12n + 10 = 0
2n2 + 12n = -10

16. Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.
The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.
2n2 + 12n = -10 2
n2 + 6n = -5
n2 + 6n + 9 =

17. Solving Quadratic Equations by Completing the Square
Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
n2 + 6n + 9 =
(n+3) (n+3) = 4
(n+3) 2 = 4

18. Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side
Step 5: Set up the two possibilities and solve
(n+3) 2 = 4
n+3 = ±2
n = -3 ± 2
n = = -1
and
n = -3 – 2 = -5