5.3Factoring Quadratic Function
12/7/2012
are the numbers you multiply together to get another number:
3 and 4 are factors of 12, because 3x4=12.2 and 5 are factors of 10, because 2x5=10
VocabularyFactors:
Examples:
Multiplying Binomials:
FOILFirst, Outside, Inside, Last
Ex. (x + 3)(x + 5) ( x + 3)( x + 5)
(x + 3)(x + 5) = x2 + 5x + 3x +15 = x2 + 8x + 15
I
O
L
First: x •x = x2
Outside: x • 5 = 5xInside: 3•x = 3xLast: 3•5 = 15
F
To multiply 2 Binomials (expression with 2 terms) use FOIL.
In this section, we’re going in reverse where the problem is Factoring x2 + 8x + 15 and your answer is (x + 3) (x + 5)
The Big “X” method
c
b
Think of 2 numbers that Multiply to c and Add to b
#1 #2
add
multiply
Answer: (x ± #1) (x ± #2)
Factor: x2 + bx + c
15
8
Think of 2 numbers that Multiply to 15 and Add to 8
3 x 5 = 153 + 5 = 8
3 5
Answer: (x + 3) (x + 5)
Factor: x2 + 8x + 15
c
b
#1 #2
add
multiply
Multiplying integersPositive X Positive = PositivePositive X Negative = NegativeNegative X Negative = Positive
Adding IntegersPositive + Positive = PositiveNegative + Negative = NegativePositive + Negative = Subtract and take sign of bigger number
Quick Review
8
-6
Think of 2 numbers that Multiply to 8 and Add to -6
-4 x -2 = 8-4 + -2 = -6-4 -2
Answer: (x - 4) (x - 2)To check: Foil (x – 4)(x – 2) and see if you get x2-6x+8
Factor: x2 - 6x + 8
c
b
#1 #2
add
multiply
-9
8
Think of 2 numbers that Multiply to -9 and Add to 8
9 x -1 = -99 + -1 = 8
-1 9
Answer: (x - 1) (x + 9)
Factor: x2 + 8x - 9
c
b
#1 #2
add
multiply
Checkpoint
Factor the expression.
Factor x 2 bx+ c+
1. x 2 6x+ 5+ ANSWER ( )1+x ( )5+x
2. b 2 7b+ 12+ ANSWER ( )3+b ( )4+b
3. s 2 5s 4+– ANSWER ( )4–s ( )1s –
ANSWER ( )12+y ( )1y –4. y 2 11y 12+ –
5. x 2 x+ 6– ANSWER ( )3+x ( )2x –
means finding the values of x that would make the equation equal to 0.
Solving ax2+bx+c = 0
Zero Product Property
When the product of two expressions equals zero , then at least one of the expressions must equal zero.
If AB = 0 , then A = 0 or B = 0 .If (x + 9)(x + 3) = 0, then x + 9 = 0 or x + 3 = 0 .
Example:
Solve the equation . - x 2 + 2x 15 = 0
SOLUTION
=x 2 + 2x 15– 0
Factor.= 0( )3x – ( )5x +
= 03x – or 5x + = 0 Use the zero product property.
= 3x x = 5– Solve for x.
ANSWER The solutions are 3 and 5.–
+ 3 = + 3 - 5 = - 5
(mult)-15
2(add)
5 -3
Solve the equation .x 2 + 9x = - 8
SOLUTION
=x 2 + 9x 8+ 0
Factor.= 0( )8x + ( )1x +
= 08x + or 1x + = 0 Use the zero product property.
= -8x x = 1– Solve for x.
ANSWER The solutions are -8 and -1.
- 8 = - 8 - 1 = - 1
(mult)8
9(add)
8 1
Rewrite in standard form
=x 2 + 9x -8
Checkpoint
ANSWER 9, 1
Solve the equation.
Solve a Quadratic Equation by Factoring
ANSWER 7, 2–
1. =x 2 10x + 9– 0
2. =y 2 5y+ 14
3. =x 2 5– 4x– ANSWER 5, 1–
Homework
5.3 p.237 #18-42even.