1. 5.4 Factoring Quadratic Expressions

2. WAYS TO SOLVE A QUADRATIC EQUATION ax² + bx + c = 0
There are many ways to solve a quadratic.
The main ones are:
Graphing
Factoring
Bottom’s Up
Grouping
Quadratic formula
Completing the square

3. By Graphing y = (x + 2)(x – 4)
By looking at the roots, we can get the solutions.
Here, the solutions are -2 and 4.

4. Golden Rules of Factoring

5. Example: Factor out the greatest common factor
4x2 + 20x -12

6. Practice: Factor each expression
Solutions:
a.) 3(3x2 + x – 6)
b) 7(p2 + 3)
c) 2w(2w + 1)
a) 9x2 + 3x – 18
b) 7p2 + 21
c) 4w2 + 2w

7. Factor Diamonds x² + 8x + 7 =0 = (x + 1) (x + 7) = 0
So your answers are -1 and -7

8. Practice: Solve by a factor diamond
X2 + 15x + 36
(x+3)(x+12)

9. Bottom’s up (Borrowing Method)
2x² + 13x + 6 =0
x² + 13x + 12 =0 12 12 1
= (x + 12) (x + 1) =0 13 2 2
= (x + 6) (x + 1) =0 2
Multiply by 2 to get rid of the fraction
= (x + 6) (2x + 1) =0
So your answers are -6 and -1/2

10. Practice: Solve using Bottom’s Up/Barrowing Method
2x2 – 19x + 24
(x-8)(2x-3)

11. Factor by Grouping 2x² – 7x – 15 =0 2x² – 10x + 3x – 15 =0
-30
-10 3 -7
Note: you are on the right track because you have (x-5) in both parenthesis
2x(x – 5) + 3(x – 5) =0
(2x + 3)(x – 5)=0
So your answers are -3/2 and 5

12. Practice: Factor by Grouping
3x2 + 7x - 20
(x+4)(3x-5)

13. SHORTCUTS a2 + 2ab + b2 (a+b)2 a2 - 2ab + b2 (a - b)2
Example: 25x2 + 90x + 81
(5x + 9)2
a2 - 2ab + b (a - b)2
Example: 9x2 – 42x + 49
(3x – 7)2
a2 - b (a+b)(a - b)
Example: x2 – 64
(x + 8)(x – 8)

14. Practice Problems: Solve using any method
Solutions:
a) (x-6)(3x+2)
(x+2)(4x-3)
(2x+7)(2x-7)
(x+4)(2x+3)
3x2 – 16x – 12
4x2 + 5x – 6
4x2 – 49
2x2 + 11X + 12