2.5 Solve Equations in Factored FormVocabularyVocabulary

Let a and b be real numbers. If ab = 0, then = 0 or = 0.

Zero-Product Property a b

2.5 Solve Equations in Factored Form

Example 1 Use the zero-product propertyUse the zero-product property

Solve 045 xxWrite original equation. Write original equation. 045 xx

______________ ______________ property. property.

Zero-product0_____ 0_____or

Solve for Solve for xx. . x = ______ or x = _____

5x 4x5 4

The solutions of the equation are ______________.5 and 4Check. Substitute each solution into the original equation to check.

04__5__?

5 5 04__5__?

4 40_____

?

90 0__ 0 0_____?

09 0__ 0

2.5 Solve Equations in Factored FormCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

036 .1 xx

06 x 03 x6x

66

3x

33

058 .2 xx

08 x 05 x8x

88

5x

55

2.5 Solve Equations in Factored Form

Binomial Trinomial 4 or more terms

Methods of Factoring

GCF GCF GCFDifference of

SquaresPerfect Square

TrinomialTwo Binomials

(Shortcut)Two Binomials

(cross)

Grouping

2.5 Solve Equations in Factored Form

Example 2 Solve an equation by factoringSolve an equation by factoring

Solve 0153 2 xxWrite original equation. Write original equation. 0153 2 xxFactor left side. (GCF) Factor left side. (GCF) 0_________ 3x x 5Zero-product property. Zero-product property. 0___ 0_____or

Solve for Solve for xx. . x = ______ or x = _____

x3 5x0 5

The solutions of the equation are ______________.0 and 5

2.5 Solve Equations in Factored Form

Example 3 Solve an equation by factoringSolve an equation by factoring

Solve bb 249 2 Write original equation. Write original equation. bb 249 2 Subtract _____ from Subtract _____ from each side. each side.

24b0249 2 bb

Factor left side. (GCF) Factor left side. (GCF) 0_________ 3b b3 8Zero-product property. Zero-product property. 0___ 0_____or

Solve for Solve for bb. . x = ______ or x = _____

b3 83 b0 3

8

The solutions of the equation are ______________.0 and

2.5 Solve Equations in Factored FormExample 3 Solve a multi-step problemSolve a multi-step problemFountain:Fountain: A fountain sprays water from the ground into the air with an initial velocity of 20 feet per second. After how many seconds does it land on the ground?

SolutionStep 1 Write a model for the water’s height above ground.

svtth 216 Vertical motion model. Vertical motion model.

____16 2 tth vv = ___ and = ___ and ss = ___. = ___. 20 200 0Simplify. Simplify. ____16 2 th t20

2.5 Solve Equations in Factored FormExample 3 Solve a multi-step problemSolve a multi-step problemFountain:Fountain: A fountain sprays water from the ground into the air with an initial velocity of 20 feet per second. After how many seconds does it land on the ground?

SolutionStep 2 Substitute __ for h. When the water lands, its height above the ground is __ feet. Solve for t.

____16__ 2 t Substitute __ for Substitute __ for hh..

______________ Factor right side. Factor right side. 00 0t20

00

4t t4 5Zero-product property. Zero-product property. 0___ 0_______or Solve for Solve for xx. . t = ______ or t = _____

t4 54 t0 25.1

The water lands on the ground ______ after it is sprayed.1.25

2.5 Solve Equations in Factored FormCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

07 Solve .3 2 dd

0 d d 70d 07 d

7d

77

2.5 Solve Equations in Factored FormCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

bb 28 Solve .4 2

0 2 b b4 102 b 014 b

1b4

11

028 2 bb

2 2

4 4b 0

b4

1

2.5 Solve Equations in Factored FormCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises.

5. In Example 4, suppose the initial velocity is 18 feet per second. After how many seconds does the water land on the ground? svtth 216

2160 t t18 0 2 t t8 9

02 t 098 t9t8

99

2 2

8 8t 0t 125.1

The water lands on the ground ______ after it is sprayed.1.125

2.5 Solve Equations in Factored Form