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

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)

× × 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)

× × × × 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

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.

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

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