chapter 6 section 5 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

Chapter Chapter 66Section Section 55

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Solving Quadratic Equations by Factoring

Solve quadratic equations by factoring.Solve other equations by factoring.

11

22

6.56.56.56.5


Solving Quadratic Equations by Factoring.

A quadratic equation is an equation that can be written in the form

ax2 + bx + c = 0,

where a, b, and c are real numbers, with a ≠ 0.

Slide 6.5 - 3

The form ax2 + bx + c = 0 is the standard form of a quadratic equation. For example,

and are all quadratic equations, but only x2 + 5x +6 = 0 is in standard form.

2 0,5 6x x 22 5 3,x x 2 4x

Until now, we have factored expressions, including many quadratic expressions. In this section we see how we can use factored quadratic expressions to solve quadratic equations.


Objective 11

Solve quadratic equations byfactoring.

Slide 6.5 - 4


We use the zero-factor property to solve a quadratic equation by factoring.

If a and b are real numbers and if ab = 0, then a = 0 or b = 0.

That is, if the product of two numbers is 0, then at least one of the numbers must be 0. One number must, but both may be 0.

Solve quadratic equations by factoring.

Slide 6.5 - 5


EXAMPLE 1

Solve. Solution:

Using the Zero-Factor Property

Slide 6.5 - 6

2 3 5 7 0x x

2 4 0x x

2 3 0x

0x

32 33 0x 2 3

2 2

x

3

2x

75 77 0x 5 7

5 5

x

7

5x

5 7 0x

2 4 0x 42 44 0x

2 4

2 2

x

2x 0, 2

3 7,

2 5

or

or


Solve.

EXAMPLE 2

Solution:

Solving Quadratic Equations

Slide 6.5 - 7

2 2 8x x

2 30x x

4 0x

6 0x

4 44 0x 4x

2 22 0x 2x

2 0x

5 0x 6 66 0x

6x 5 55 0x

5,6

2 8 82 8x x 2 2 8 0x x

4 2 0x x 4,2

2 3 30030x xx x 2 30 0x x

6 5 0x x

5x

or

or


Solve quadratic equations by factoring. (cont’d)

Slide 6.5 - 8

In summary, follow these steps to solve quadratic equations by factoring.

Step 1: Write the equation in standard form— that is, with all terms on one side of the equals sign in descending power of the variable and 0 on the other side.

Step 2: Factor completely.

Step 3: Use the zero-factor property to set each factor with variable equal to 0, and solve the resulting equations.

Step 4: Check each solution in the original equation.


EXAMPLE 3

Solution:

Solving a Quadratic Equation with a Common Factor

Slide 6.5 - 9

Solve 3m2 − 9m = 30.

2 303 9 3 300m m

53 2 0m m

23 9 30 0m m

2 3 10 03 m m

2,5

2 0m 2 22 0m

2m

5 0m

5 55 0m

5m

A common error is to include the common factor 3 as a solution. Only factors containing variables lead to solutions.


EXAMPLE 4

Solution:

249 9 0x

7 3 7 3 0x x

2 3x x

3 3,

7 7

3 33 0x

Solving Quadratic Equations

Slide 6.5 - 10

Solve.

0,3

2 333x xx x 3 0x x

0x

3x

37 33 0x

3

7x

7 3

7 7

x

37 33 0x 7 3

7 7

x

3

7x


EXAMPLE 4

Solution:

Solving Quadratic Equations (cont’d)

Slide 6.5 - 11

Solve.

4 7 2x x

24 2 27 2x x 24 7 2 0x x

2 4 1 0x x

12,

4

2 22 0x 2x

2 0x

4 1

4 4

x

14 11 0x

1

4x

4 1 0x


Objective 22

Solve other equations by factoring.

Slide 6.5 - 12


EXAMPLE 5

Solve.

Solution:

32 50 0x x

Solving Equations with More than Two Variable Factors

Slide 6.5 - 13

22 25 0x x

2 5 5 0x x x

2 0x 2 0

2 2

x

0x 0, 5,5

5 55 0x 5x

5 55 0x 5x


EXAMPLE 5

Solve.

Solving Equations with More than Two Variable Factors (cont’d)

Slide 6.5 - 14

22 1 2 7 15 0x x x

2 1 2 3 5 0x x x

12 11 0x 2 1

2 2

x

1

2x

32 33 0x 2 3

2 2

x

3

2x

5 55 0x

1 35, ,

2 2

5x

Solution:


Solve.

EXAMPLE 6

21 2 1 1x x x

Solving an Equation Requiring Multiplication before Factoring

Slide 6.5 - 15

2 22 3 1 2 1x x x x 2 22 22 1 2 12 3 1 2 1x x x xx x x x

2 5 0x x 5 0x x

0x 5 55 0x 5x 0,5

Solution:

chapter 6 section 5 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

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