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.