1. Factoring Quadratic Expression
Today’s Objective:
I can factor a quadratic expression.

2. Multiplying Binomial factors:
numbers/expressions that have a product equal to the given number/expression
Factors:
Multiplying Binomial factors:
Factoring 𝑎 𝑥 2 +𝑏𝑥+𝑐 when 𝑎=1
FOIL
𝑥 2 +9𝑥+20 × ×
Factors of c (20): Add to b (9)
1, 20
2, 10
4, 5
(𝑥+3)(𝑥+5) 21 12 9
First
(𝑥+4)(𝑥+5)
Outer
Inner
Last
Tips
c is negative
factors are opposite
b is positive
larger factor is positive
𝑥 𝑥
+𝑥 5
+3(𝑥)
+3(5)
𝑥 2 +2𝑥−15 ×
𝑥 2 +8𝑥+15
Factors of c (-15): Add to b (2)
−1, 15
−3, 5 14 2
(2𝑥−3)(𝑥+7)
(𝑥−3)(𝑥+5)
2𝑥 2 +11𝑥−21
𝑥 2 −11𝑥+30
(𝑥−5)(𝑥−6)

3. × × Factoring common factors Factoring 𝑎 𝑥 2 +𝑏𝑥+𝑐 when 𝑎≠1 6𝑥 2 +9𝑥
=3𝑥( ) 2𝑥 +3
2 𝑥 2 +11𝑥+12 3𝑥 3𝑥 × ×
Factors of ac (24): Add to b (11)
1, 24
2, 12
3, 8
7𝑥 2 −21
=7 ( )
𝑥 2 −3 25 14 11 7 7
2 𝑥 2 +3𝑥+8𝑥+12 𝑥 𝑥 4 4
4𝑥 2 +20𝑥−56
𝑥 ( ) 4 4 4
2𝑥+3
+4 ( )
2𝑥+3
=4 ( )
𝑥 2
+5𝑥
−14
(2𝑥+3)(𝑥+4)
=4(𝑥+7)(𝑥−2)
−𝑥 2 +14𝑥+32
=−1 ( )
𝑥 2 −14𝑥−32
=−1(𝑥+2)(𝑥−16)

4. × × × × Factoring 𝑎 𝑥 2 +𝑏𝑥+𝑐 when 𝑎≠1 3 𝑥 2 −11𝑥−20 4 𝑥 2 −4𝑥−3
Factors of ac (-60): Add to b (-11)
1, −60
2, −30
3, −20
4, −15
Factors of ac (-12): Add to b (-4)
1, −12
2, −6
−59
−28
−17
−11
−11 −4
3 𝑥 2 +4𝑥−15𝑥−20
4 𝑥 2 +2𝑥−6𝑥−3 𝑥 𝑥 −5 −5 2𝑥 2𝑥 −3 −3
𝑥(3𝑥+4)
−5 (3𝑥+4)
2𝑥(2𝑥+1)
−3 (2𝑥+1)
(3𝑥+4)(𝑥−5)
(2𝑥+1)(2𝑥−3)
Tips for factoring
Factor common factors first
If possible have 𝑎=1

5. For any real numbers a and b, if ab=0, then either a=0, b=0, or both.
Zero Product Property
For any real numbers a and b, if ab=0, then either a=0, b=0, or both.

6. In the next example, you must set the equation equal to zero before factoring. Then set the individual factors equal to zero and solve.
p. 221:15-31 odd
p. 229: 9-17 odd evens

7. p. 221:15-31 odd
p. 229: 9-17 odd