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OBJECTIVE:

1. Solve an equation with fractions by writing it in quadratic form.

2. Use quadratic equations to solve applied problems.

3. Solve an equation with radicals by writing it in quadratic form.

4. Solve an equation that is quadratic in form by substitution.

1. Solve an equation with fractions by writing it in quadratic form.

2. Use quadratic equations to solve applied problems.

3. Solve an equation with radicals by writing it in quadratic form.

4. Solve an equation that is quadratic in form by substitution.

1.
Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 10.3 - 1

Sec 10.3 - 1

2.
Chapter 10

Quadratic Equations,

Inequalities,

and Functions

Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 10.3 - 2

Quadratic Equations,

Inequalities,

and Functions

Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 10.3 - 2

3.
10.3

Equations Quadratic in Form

Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 10.3 - 3

Equations Quadratic in Form

Copyright © 2010 Pearson Education, Inc. All rights reserved

Sec 10.3 - 3

4.
10.3 Equations Quadratic in Form

Objectives

1. Solve an equation with fractions by writing it in quadratic

form.

2. Use quadratic equations to solve applied problems.

3. Solve an equation with radicals by writing it in quadratic

form.

4. Solve an equation that is quadratic in form by substitution.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 4

Objectives

1. Solve an equation with fractions by writing it in quadratic

form.

2. Use quadratic equations to solve applied problems.

3. Solve an equation with radicals by writing it in quadratic

form.

4. Solve an equation that is quadratic in form by substitution.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 4

5.
10.3 Equations Quadratic in Form

EXAMPLE 1 Solving an Equation with Fractions That

Leads to a Quadratic Equation

Solve 3 2 5 .

+ =

x x–5 6

Clear fractions by multiplying each term by the least common denominator,

6x(x – 5). (Note that the domain must be restricted to x ≠ 0 and x ≠ 5.)

3 2 5

6x(x – 5) + 6x(x – 5) = 6x(x – 5)

x x–5 6

18(x – 5) + 12x = 5x(x – 5)

18x – 90 + 12x = 5x2 – 25x Distributive property

30x – 90 = 5x2 – 25x Combine terms.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 5

EXAMPLE 1 Solving an Equation with Fractions That

Leads to a Quadratic Equation

Solve 3 2 5 .

+ =

x x–5 6

Clear fractions by multiplying each term by the least common denominator,

6x(x – 5). (Note that the domain must be restricted to x ≠ 0 and x ≠ 5.)

3 2 5

6x(x – 5) + 6x(x – 5) = 6x(x – 5)

x x–5 6

18(x – 5) + 12x = 5x(x – 5)

18x – 90 + 12x = 5x2 – 25x Distributive property

30x – 90 = 5x2 – 25x Combine terms.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 5

6.
10.3 Equations Quadratic in Form

EXAMPLE 1 Solving an Equation with Fractions That

Leads to a Quadratic Equation

Solve 3 2 5 .

+ =

x x–5 6

Combine and rearrange terms so that the quadratic equation is in standard

form. Then factor to solve the resulting equation.

30x – 90 = 5x2 – 25x

5x2 – 55x + 90 = 0 Standard form

5(x2 – 11x + 18) = 0 Factor.

5(x – 2)(x – 9) = 0 Factor.

x – 2 = 0 or x – 9 = 0 Zero-factor property.

x = 2 or x = 9 Solve each equation.

Check by substituting these solutions in the original equation. The solution

set is { 2, 9 }.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 6

EXAMPLE 1 Solving an Equation with Fractions That

Leads to a Quadratic Equation

Solve 3 2 5 .

+ =

x x–5 6

Combine and rearrange terms so that the quadratic equation is in standard

form. Then factor to solve the resulting equation.

30x – 90 = 5x2 – 25x

5x2 – 55x + 90 = 0 Standard form

5(x2 – 11x + 18) = 0 Factor.

5(x – 2)(x – 9) = 0 Factor.

x – 2 = 0 or x – 9 = 0 Zero-factor property.

x = 2 or x = 9 Solve each equation.

Check by substituting these solutions in the original equation. The solution

set is { 2, 9 }.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 6

7.
10.3 Equations Quadratic in Form

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 1 Read the problem carefully.

Step 2 Assign a variable. Let x = the speed of the current. The current

slows down the boat when it is going upstream, so the rate (or

speed) upstream is the speed of the boat in still water less the sp

of the current, or 10 – x.

Riverboat traveling

upstream – the current

slows it down.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 7

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 1 Read the problem carefully.

Step 2 Assign a variable. Let x = the speed of the current. The current

slows down the boat when it is going upstream, so the rate (or

speed) upstream is the speed of the boat in still water less the sp

of the current, or 10 – x.

Riverboat traveling

upstream – the current

slows it down.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 7

8.
10.3 Equations Quadratic in Form

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 2 Assign a variable. Let x = the speed of the current. The current

slows down the boat when it is going upstream, so the rate (or

speed) upstream is the speed of the boat in still water less the sp

of the current, or 10 – x.

Similarly, the current speeds up the

boat as it travels downstream, so

its speed downstream is 10 + x.

Thus,

10 – x = the rate upstream;

10 + x = the rate downstream.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 8

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 2 Assign a variable. Let x = the speed of the current. The current

slows down the boat when it is going upstream, so the rate (or

speed) upstream is the speed of the boat in still water less the sp

of the current, or 10 – x.

Similarly, the current speeds up the

boat as it travels downstream, so

its speed downstream is 10 + x.

Thus,

10 – x = the rate upstream;

10 + x = the rate downstream.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 8

9.
10.3 Equations Quadratic in Form

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 2 Assign a variable. Let x = the speed of the current. The current

slows down the boat when it is going upstream, so the rate (or

speed) upstream is the speed of the boat in still water less the sp

of the current, or 10 – x.

d r t

Upstream 8 10 – x 8

10 – x

8

Downstream 8 10 + x 10 + x

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 9

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 2 Assign a variable. Let x = the speed of the current. The current

slows down the boat when it is going upstream, so the rate (or

speed) upstream is the speed of the boat in still water less the sp

of the current, or 10 – x.

d r t

Upstream 8 10 – x 8

10 – x

8

Downstream 8 10 + x 10 + x

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 9

10.
10.3 Equations Quadratic in Form

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 3 Write an equation. The total time, 1 hr and 40 min, can be written as

1 + 40 =+ 1 + 2 = = 5

60 3 3 hr.

Time upstream Time downstream Total Time

d r t

Upstream 8 10 – x 8

10 – x

8

Downstream 8 10 + x 10 + x

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 10

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 3 Write an equation. The total time, 1 hr and 40 min, can be written as

1 + 40 =+ 1 + 2 = = 5

60 3 3 hr.

Time upstream Time downstream Total Time

d r t

Upstream 8 10 – x 8

10 – x

8

Downstream 8 10 + x 10 + x

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 10

11.
10.3 Equations Quadratic in Form

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 4 Solve the equation. Multiply each side by 3(10 – x)(10 + x), the LCD,

and solve the resulting quadratic equation.

8 + 8 = 5

10 – x 10 + x 3

3(10 + x)8 + 3(10 – x)8 = 5(10 – x)(10 + x)

24(10 + x) + 24(10 – x) = 5(100 – x2)

240 + 24x + 240 – 24x = 500 – 5x2 Distributive property

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 11

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 4 Solve the equation. Multiply each side by 3(10 – x)(10 + x), the LCD,

and solve the resulting quadratic equation.

8 + 8 = 5

10 – x 10 + x 3

3(10 + x)8 + 3(10 – x)8 = 5(10 – x)(10 + x)

24(10 + x) + 24(10 – x) = 5(100 – x2)

240 + 24x + 240 – 24x = 500 – 5x2 Distributive property

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 11

12.
10.3 Equations Quadratic in Form

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 4 Solve the equation. Multiply each side by 3(10 – x)(10 + x), the LCD,

and solve resulting quadratic equation.

240 + 24x + 240 – 24x = 500 – 5x2

480 = 500 – 5x2 Combine terms.

5x2 = 20

x2 = 4 Divide by 5.

x = 2 or x = –2 Square root property

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 12

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 4 Solve the equation. Multiply each side by 3(10 – x)(10 + x), the LCD,

and solve resulting quadratic equation.

240 + 24x + 240 – 24x = 500 – 5x2

480 = 500 – 5x2 Combine terms.

5x2 = 20

x2 = 4 Divide by 5.

x = 2 or x = –2 Square root property

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 12

13.
10.3 Equations Quadratic in Form

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 5 State the answer. The speed of the current cannot be –2, so the

answer is 2 mph.

Step 6 Check that this value satisfies the original problem.

d r t

Upstream 8 10 8

–2x 1

Downstream 8 1012

+2x 2

3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 13

EXAMPLE 2 Solving a Motion Problem

A riverboat for tourists averages 10 mph in still water. It takes the boat 1 hr,

40 minutes to go 8 miles upstream and return. Find the speed of the current.

Step 5 State the answer. The speed of the current cannot be –2, so the

answer is 2 mph.

Step 6 Check that this value satisfies the original problem.

d r t

Upstream 8 10 8

–2x 1

Downstream 8 1012

+2x 2

3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 13

14.
10.3 Equations Quadratic in Form

Caution on “Solutions”

As shown in Example 2, when a quadratic equation is used to solve an

applied problem, sometimes only one answer satisfies the application.

Always check each answer in the words of the original problem.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 14

Caution on “Solutions”

As shown in Example 2, when a quadratic equation is used to solve an

applied problem, sometimes only one answer satisfies the application.

Always check each answer in the words of the original problem.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 14

15.
10.3 Equations Quadratic in Form

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 1 Read the problem again. There will be two answers.

Step 2 Assign a variable. Let x represent the number of hours for the

slower carpet layer to complete the job alone. Then the faster car

layer could do the entire job in (x – 2) hours.

1 is

The slower person’s rate , and the faster person’s rate1is

x x–2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 15

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 1 Read the problem again. There will be two answers.

Step 2 Assign a variable. Let x represent the number of hours for the

slower carpet layer to complete the job alone. Then the faster car

layer could do the entire job in (x – 2) hours.

1 is

The slower person’s rate , and the faster person’s rate1is

x x–2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 15

16.
10.3 Equations Quadratic in Form

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 2 (continued) Now complete the table below.

1 is

The slower person’s rate , and the faster person’s rate1is

x x–2

Together, they can do the job in 5 hr.

Time Working Fractional Par

Rate Together of the Job Don

1 1 (5)

Slower Worker x 5 x

1 5 1 (5)

Faster Worker x–2 x–2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 16

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 2 (continued) Now complete the table below.

1 is

The slower person’s rate , and the faster person’s rate1is

x x–2

Together, they can do the job in 5 hr.

Time Working Fractional Par

Rate Together of the Job Don

1 1 (5)

Slower Worker x 5 x

1 5 1 (5)

Faster Worker x–2 x–2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 16

17.
10.3 Equations Quadratic in Form

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 3 Write an equation. The sum of the fractional parts done by the

workers should equal 1 (the whole job).

Part done by Part done by

+ = 1 whole job.

slower worker faster worker

5 + 5 = 1

x x–2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 17

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 3 Write an equation. The sum of the fractional parts done by the

workers should equal 1 (the whole job).

Part done by Part done by

+ = 1 whole job.

slower worker faster worker

5 + 5 = 1

x x–2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 17

18.
10.3 Equations Quadratic in Form

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 4 Solve the equation from Step 3.

5 + 5 = 1

x x–2

x(x – 2) 5 + 5 = x(x – 2) 1 Multiply by the LCD.

x x–2

5(x – 2) + 5x = x(x – 2) Distributive property

5x – 10 + 5x = x2 – 2x Distributive property

0 = x2 – 12x + 10 Standard form

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 18

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 4 Solve the equation from Step 3.

5 + 5 = 1

x x–2

x(x – 2) 5 + 5 = x(x – 2) 1 Multiply by the LCD.

x x–2

5(x – 2) + 5x = x(x – 2) Distributive property

5x – 10 + 5x = x2 – 2x Distributive property

0 = x2 – 12x + 10 Standard form

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 18

19.
10.3 Equations Quadratic in Form

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 4 Solve the equation. (continued)

0 = x2 – 12x + 10

This equation cannot be solved by factoring, so use the quadratic

formula. (a = 1, b = –12, c = 10)

x = –b + b2 – 4ac x =

12 + 144 – 40

2a 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 19

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 4 Solve the equation. (continued)

0 = x2 – 12x + 10

This equation cannot be solved by factoring, so use the quadratic

formula. (a = 1, b = –12, c = 10)

x = –b + b2 – 4ac x =

12 + 144 – 40

2a 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 19

20.
10.3 Equations Quadratic in Form

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 4 Solve the equation. (continued)

12 + 144 – 40 12 – 144 – 40

x = or x =

2 2

x ≈ 11.1 or x ≈ .9

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 20

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 4 Solve the equation. (continued)

12 + 144 – 40 12 – 144 – 40

x = or x =

2 2

x ≈ 11.1 or x ≈ .9

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 20

21.
10.3 Equations Quadratic in Form

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 5 State the answer. Only the solution 11.1 makes sense in the

original problem. (Why?) Thus, the slower worker can do the job i

about 11.1 hr and the faster in about 11.1 – 2 = 9.1 hr.

Step 6 Check that these results satisfy the original problem.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 21

EXAMPLE 3 Solving a Work Problem

It takes two carpet layers 5 hr to carpet a room. If each worked alone, one of

them could do the job in 2 hr less time than the other. How long would it take

each carpet layer to complete the job alone?

Step 5 State the answer. Only the solution 11.1 makes sense in the

original problem. (Why?) Thus, the slower worker can do the job i

about 11.1 hr and the faster in about 11.1 – 2 = 9.1 hr.

Step 6 Check that these results satisfy the original problem.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 21

22.
10.3 Equations Quadratic in Form

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(a) n = –2n + 15

This equation is not quadratic. However, squaring both sides of the

equation gives a quadratic equation that can be solved by factoring.

n2 = –2n + 15 Square both side

n2 + 2n – 15 = 0 Standard form

(n + 5)(n – 3) = 0 Factor.

n + 5 = 0 or n – 3 = 0 Zero-factor property

n = –5 or n = 3 Potential solution

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 22

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(a) n = –2n + 15

This equation is not quadratic. However, squaring both sides of the

equation gives a quadratic equation that can be solved by factoring.

n2 = –2n + 15 Square both side

n2 + 2n – 15 = 0 Standard form

(n + 5)(n – 3) = 0 Factor.

n + 5 = 0 or n – 3 = 0 Zero-factor property

n = –5 or n = 3 Potential solution

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 22

23.
10.3 Equations Quadratic in Form

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(a) n = –2n + 15

Recall from Section 9.6 that squaring both sides of a radical equation can

introduce extraneous solutions that do not satisfy the original equation.

All potential solutions must be checked in the original (not the

squared) equation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 23

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(a) n = –2n + 15

Recall from Section 9.6 that squaring both sides of a radical equation can

introduce extraneous solutions that do not satisfy the original equation.

All potential solutions must be checked in the original (not the

squared) equation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 23

24.
10.3 Equations Quadratic in Form

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(a) n = –2n + 15

Check: If n = –5, then If n = 3, then

n = –2n + 15 n = –2n + 15

–5 = –2(–5) + 15 ? 3 = –2(3) + 15 ?

–5 = 25 3 = 9

–5 = 5 False 3 = 3 True

Only the solution 3 checks, so the solution set is { 3 }.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 24

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(a) n = –2n + 15

Check: If n = –5, then If n = 3, then

n = –2n + 15 n = –2n + 15

–5 = –2(–5) + 15 ? 3 = –2(3) + 15 ?

–5 = 25 3 = 9

–5 = 5 False 3 = 3 True

Only the solution 3 checks, so the solution set is { 3 }.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 24

25.
10.3 Equations Quadratic in Form

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(b) 3 e + e = 10

3 e = 10 – e Isolate the radical on one side.

9e = 100 – 20e + e2 Square both sides.

0 = e2 – 29e + 100 Standard form

0 = (e – 4)(e – 25) Factor.

e – 4 = 0 or e – 25 = 0 Zero-factor property

e = 4 or e = 25 Potential solutions

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 25

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(b) 3 e + e = 10

3 e = 10 – e Isolate the radical on one side.

9e = 100 – 20e + e2 Square both sides.

0 = e2 – 29e + 100 Standard form

0 = (e – 4)(e – 25) Factor.

e – 4 = 0 or e – 25 = 0 Zero-factor property

e = 4 or e = 25 Potential solutions

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 25

26.
10.3 Equations Quadratic in Form

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(b) 3 e + e = 10

Check both potential solutions, 4 and 25, in the original equation.

Check: If e = 4, then If e = 25, then

3 e + e = 10 3 e + e = 10

3 4 + 4 = 10 ? 3 25 + 25 = 10 ?

6 + 4 = 10 ? 15 + 25 = 10 ?

10 = 10 True 40 = 10 False

Only the solution 4 checks, so the solution set is { 4 }.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 26

EXAMPLE 4 Solving Radical Equations That Lead to

Quadratic Equations

Solve each equation.

(b) 3 e + e = 10

Check both potential solutions, 4 and 25, in the original equation.

Check: If e = 4, then If e = 25, then

3 e + e = 10 3 e + e = 10

3 4 + 4 = 10 ? 3 25 + 25 = 10 ?

6 + 4 = 10 ? 15 + 25 = 10 ?

10 = 10 True 40 = 10 False

Only the solution 4 checks, so the solution set is { 4 }.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 26

27.
10.3 Equations Quadratic in Form

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(a) m4 – 26m2 + 25 = 0.

Because m4 = (m2) 2, we can write this equation in quadratic form with

u = m2 and u2 = m4. (Instead of u, any letter other than m could be used.)

m4 – 26m2 + 25 = 0

(m2)2 – 26m2 + 25 = 0 m4 = (m2)2

u2 – 26u + 25 = 0 Let u = m2.

(u – 1)(u – 25) = 0 Factor.

u – 1 = 0 or u – 25 = 0 Zero-factor property

u = 1 or u = 25 Solve.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 27

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(a) m4 – 26m2 + 25 = 0.

Because m4 = (m2) 2, we can write this equation in quadratic form with

u = m2 and u2 = m4. (Instead of u, any letter other than m could be used.)

m4 – 26m2 + 25 = 0

(m2)2 – 26m2 + 25 = 0 m4 = (m2)2

u2 – 26u + 25 = 0 Let u = m2.

(u – 1)(u – 25) = 0 Factor.

u – 1 = 0 or u – 25 = 0 Zero-factor property

u = 1 or u = 25 Solve.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 27

28.
10.3 Equations Quadratic in Form

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(a) m4 – 26m2 + 25 = 0.

To find m, we substitute m2 for u.

u = 1 or u = 25

m2 = 1 or m2 = 25

m = +1 or m = +5 Square root property

The equation m4 – 26m2 + 25 = 0, a fourth-degree equation, has

four solutions. * The solution set is { –5, –1, 1, 5 }. Check by substitution.

* In general, an equation in which an nth-degree polynomial equals 0 has

n solutions, although some of them may be repeated.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 28

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(a) m4 – 26m2 + 25 = 0.

To find m, we substitute m2 for u.

u = 1 or u = 25

m2 = 1 or m2 = 25

m = +1 or m = +5 Square root property

The equation m4 – 26m2 + 25 = 0, a fourth-degree equation, has

four solutions. * The solution set is { –5, –1, 1, 5 }. Check by substitution.

* In general, an equation in which an nth-degree polynomial equals 0 has

n solutions, although some of them may be repeated.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 28

29.
10.3 Equations Quadratic in Form

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(b) c4 = 10c2 – 2.

First write the equation as

c4 – 10c2 + 2 = 0 or ( c2 )2 – 10c2 + 2 = 0,

which is quadratic in form with u = c2. Substitute u for c2 and u2 for c4 to get

u2 – 10u + 2 = 0.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 29

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(b) c4 = 10c2 – 2.

First write the equation as

c4 – 10c2 + 2 = 0 or ( c2 )2 – 10c2 + 2 = 0,

which is quadratic in form with u = c2. Substitute u for c2 and u2 for c4 to get

u2 – 10u + 2 = 0.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 29

30.
10.3 Equations Quadratic in Form

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(b) 4 2 x = –b + b2 – 4ac

c = 10c – 2. 2a

Since this equation cannot be solved by factoring, use the quadratic formula.

u2 – 10u + 2 = 0

10 + 100 – 8

u = a = 1, b = –10, c = 2

2

10 + 92

u =

2

10 + 2 23

u = 92 = 4 · 23 = 2 23

2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 30

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(b) 4 2 x = –b + b2 – 4ac

c = 10c – 2. 2a

Since this equation cannot be solved by factoring, use the quadratic formula.

u2 – 10u + 2 = 0

10 + 100 – 8

u = a = 1, b = –10, c = 2

2

10 + 92

u =

2

10 + 2 23

u = 92 = 4 · 23 = 2 23

2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 30

31.
10.3 Equations Quadratic in Form

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(b) c4 = 10c2 – 2.

2 5 + 23

u = Factor.

2

u = 5 + 23 Lowest terms

c2 = 5 + 23 or c2 = 5 – 23 Substitute c2 for u.

c = + 5 + 23 or c = + 5 – 23 Square root property

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 31

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(b) c4 = 10c2 – 2.

2 5 + 23

u = Factor.

2

u = 5 + 23 Lowest terms

c2 = 5 + 23 or c2 = 5 – 23 Substitute c2 for u.

c = + 5 + 23 or c = + 5 – 23 Square root property

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 31

32.
10.3 Equations Quadratic in Form

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(b) c4 = 10c2 – 2.

The solution set contains four numbers:

5 + 23 , – 5 + 23 , 5 – 23 – , 5 – 23

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 32

EXAMPLE 6 Solving Equations That are Quadratic

in form

Solve each equation.

(b) c4 = 10c2 – 2.

The solution set contains four numbers:

5 + 23 , – 5 + 23 , 5 – 23 – , 5 – 23

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 32

33.
10.3 Equations Quadratic in Form

Note on Solving Equations

Some students prefer to solve equations like those in Example 6 (a) by

factoring directly. For example,

m4 – 26m2 + 25 = 0 Example 6(a) equation

(m2 – 1)(m2 – 25) = 0 Factor.

(m + 1)(m – 1)(m + 25)(m – 25) = 0. Factor again.

Using the zero-factor property gives the same solutions obtained in

Example 6(a). Equations that cannot be solved by factoring (as in

Example 6(c)) must be solved by substitution and the quadratic formula.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 33

Note on Solving Equations

Some students prefer to solve equations like those in Example 6 (a) by

factoring directly. For example,

m4 – 26m2 + 25 = 0 Example 6(a) equation

(m2 – 1)(m2 – 25) = 0 Factor.

(m + 1)(m – 1)(m + 25)(m – 25) = 0. Factor again.

Using the zero-factor property gives the same solutions obtained in

Example 6(a). Equations that cannot be solved by factoring (as in

Example 6(c)) must be solved by substitution and the quadratic formula.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 33

34.
10.3 Equations Quadratic in Form

EXAMPLE 7 Solving Equations That are Quadratic

in form

Solve 3(2x – 1)2 + 8(2x – 1) – 35 = 0.

Because of the repeated quantity 2x – 1, this equation is quadratic in

form with u = 2x – 1.

3(2x – 1)2 + 8(2x – 1) – 35 = 0

3u2 + 8u – 35 = 0 Let 2x – 1 = u.

(3u – 7)(u + 5) = 0 Factor.

3u – 7 = 0 or u + 5 = 0 Zero-factor property

7

u = or u = –5 Zero-factor property

3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 34

EXAMPLE 7 Solving Equations That are Quadratic

in form

Solve 3(2x – 1)2 + 8(2x – 1) – 35 = 0.

Because of the repeated quantity 2x – 1, this equation is quadratic in

form with u = 2x – 1.

3(2x – 1)2 + 8(2x – 1) – 35 = 0

3u2 + 8u – 35 = 0 Let 2x – 1 = u.

(3u – 7)(u + 5) = 0 Factor.

3u – 7 = 0 or u + 5 = 0 Zero-factor property

7

u = or u = –5 Zero-factor property

3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 34

35.
10.3 Equations Quadratic in Form

EXAMPLE 7 Solving Equations That are Quadratic

in form

Solve 3(2x – 1)2 + 8(2x – 1) – 35 = 0.

7

u = 3 or u = –5

7

2x – 1 = 3 or 2x – 1 = –5 Substitute 2x – 1 for u.

10

2x = or 2x = –4 Solve for x.

3

5

x = or x = –2 Solve for x.

3

5

Check that the solution set of the original equation is –2, .

3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 35

EXAMPLE 7 Solving Equations That are Quadratic

in form

Solve 3(2x – 1)2 + 8(2x – 1) – 35 = 0.

7

u = 3 or u = –5

7

2x – 1 = 3 or 2x – 1 = –5 Substitute 2x – 1 for u.

10

2x = or 2x = –4 Solve for x.

3

5

x = or x = –2 Solve for x.

3

5

Check that the solution set of the original equation is –2, .

3

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 35

36.
10.3 Equations Quadratic in Form

Caution

A common error when solving problems like those in Examples 6 and 7 is to

stop too soon. Once you have solved for u, remember to substitute and

solve for the values of the original variable.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 36

Caution

A common error when solving problems like those in Examples 6 and 7 is to

stop too soon. Once you have solved for u, remember to substitute and

solve for the values of the original variable.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.3 - 36