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5: Solving Problems Using Quadratic Equations

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10.5 • Solving Problems Using Quadratic Equations

441

Suggestions for Solving Word Problems

1. Read the problem carefully and make certain that you understand the meanings of all the

words. Be especially alert for any technical terms used in the statement of the problem.

2. Read the problem a second time (perhaps even a third time) to get an overview of the

situation being described and to determine the known facts as well as what is to be found.

3. Sketch any figure, diagram, or chart that might be helpful in analyzing the problem.

*4. Choose meaningful variables to represent the unknown quantities. Use one or two

variables, whichever seems easiest. The term “meaningful” refers to the choice of

letters to use as variables. Choose letters that have some significance for the problem

under consideration. For example, if the problem deals with the length and width of

a rectangle, then l and w are natural choices for the variables.

*5. Look for guidelines that you can use to help set up equations. A guideline might be

a formula such as area of a rectangular region equals length times width, or a

statement of a relationship such as the product of the two numbers is 98.

*6. (a) Form an equation containing the variable, which translates the conditions of the

guideline from English into algebra; or

(b) Form two equations containing the two variables, which translate the guidelines

from English into algebra.

*7. Solve the equation (system of equations) and use the solution (solutions) to

determine all facts requested in the problem.

8. Check all answers back in the original statement of the problem.

The asterisks indicate those suggestions that have been revised to include using systems of

equations to solve problems. Keep these suggestions in mind as you study the examples and

work the problems in this section.

Classroom Example

The length of a rectangular region is

8 inches more than its width. The

area of the region is 48 square inches.

Find the length and width of the

rectangle.

EXAMPLE 1

The length of a rectangular region is 2 centimeters more than its width. The area of the

region is 35 square centimeters. Find the length and width of the rectangle.

Solution

We let l represent the length, and we let w represent the width (see Figure 10.10). We can

use the area formula for a rectangle, A ϭ lw, and the statement “the length of a rectangular

region is 2 centimeters greater than its width” as guidelines to form a system of equations.

Area is 35 cm2.

l

Figure 10.10

w

a

lw ϭ 35

b

lϭwϩ2

The second equation indicates that we can substitute w ϩ 2 for l. Making this substitution in

the first equation yields

1w ϩ 221w2 ϭ 35

Solving this quadratic equation by factoring, we get

w2 ϩ 2w ϭ 35

w2 ϩ 2w Ϫ 35 ϭ 0

1w ϩ 721w Ϫ 52 ϭ 0

wϩ7ϭ0

or

wϪ5ϭ0

w ϭ -7

or

wϭ5

442

The width of a rectangle cannot be a negative number, so we discard the solution -7. Thus

the width of the rectangle is 5 centimeters and the length (w ϩ 2) is 7 centimeters.

Classroom Example

Find two consecutive whole numbers

whose product is 702.

EXAMPLE 2

Find two consecutive whole numbers whose product is 506.

Solution

We let n represent the smaller whole number. Then n ϩ 1 represents the next larger whole

number. The phrase “whose product is 506” translates into the equation

n1n ϩ 12 ϭ 506

Changing this quadratic equation into standard form produces

n2 ϩ n ϭ 506

n2 ϩ n Ϫ 506 ϭ 0

Because of the size of the constant term, let’s not try to factor; instead, we can use the

-1 Ϯ 212 Ϫ 41121 -5062

2112

-1 Ϯ 22025

2

-1 Ϯ 45

22025 ϭ 45

2

-1 ϩ 45

-1 Ϫ 45

or

2

2

or

n ϭ -23

n ϭ 22

Since we are looking for whole numbers, we discard the solution Ϫ23. Therefore, the whole

numbers are 22 and 23.

Classroom Example

The perimeter of a rectangular lot is

122 feet, and its area is 888 square

feet. Find the length and width of

the lot.

EXAMPLE 3

The perimeter of a rectangular lot is 100 meters, and its area is 616 square meters. Find the

length and width of the lot.

Solution

We let l represent the length, and we let w represent

the width (see Figure 10.11).

Then

a

lw ϭ 616

b

2l ϩ 2w ϭ 100

2

Area is 616 m

Perimeter is 100 m

Multiplying the second equation by

Area is

616 m 2.

Perimeter is

100 m.

1

produces

2

l ϩ w ϭ 50, which can be changed to l ϭ 50 Ϫ w.

Substituting 50 Ϫ w for l in the first equation produces

l

Figure 10.11

w

10.5 • Solving Problems Using Quadratic Equations

443

150 Ϫ w21w2 ϭ 616

50w Ϫ w2 ϭ 616

w2 Ϫ 50w ϭ - 616

Using the method of completing the square, we have

w2 Ϫ 50w ϩ 625 ϭ -616 ϩ 625

1w Ϫ 2522 ϭ 9

w Ϫ 25 ϭ Ϯ3

w Ϫ 25 ϭ 3

w ϭ 28

or

w Ϫ 25 ϭ -3

w ϭ 22

or

If w ϭ 28, then l ϭ 50 Ϫ w ϭ 22. If w ϭ 22, then l ϭ 50 Ϫ w ϭ 28. The rectangle is 28 meters

by 22 meters or 22 meters by 28 meters.

Classroom Example

Find two numbers such that their sum

is 4 and their product is 2.

Find two numbers such that their sum is 2, and their product is Ϫ1.

EXAMPLE 4

Solution

We let n represent one of the numbers, and we let m represent the other number.

a

nϩmϭ2

b

nm ϭ -1

Their sum is 2

Their product is Ϫ1

We can change the first equation to m ϭ 2 Ϫ n; then we can substitute 2 Ϫ n for m in the second equation.

n 12 Ϫ n2 ϭ - 1

2n Ϫ n2 ϭ - 1

2

-n ϩ 2n ϩ 1 ϭ 0

n2 Ϫ 2n Ϫ 1 ϭ 0

Multiply both sides by Ϫ1

-1 -22 Ϯ 21- 22 Ϫ 41121 -12

2

2112

2 Ϯ 28

2 Ϯ 222

ϭ

ϭ 1 Ϯ 22

2

2

If n ϭ 1 ϩ 22, then m ϭ 2 Ϫ 11 ϩ 222

ϭ 2 Ϫ 1 Ϫ 22

ϭ 1 Ϫ 22

If n ϭ 1 Ϫ 22, then m ϭ 2 Ϫ 11 Ϫ 222

ϭ 2 Ϫ 1 ϩ 22

ϭ 1 ϩ 22

The numbers are 1 ϩ 22 and 1 Ϫ 22 .

Classroom Example

Lynn drove 201 miles in 1 hour less

time than it took Michelle to drive

256 miles. Lynn drove at an average

rate of 3 miles per hour faster than

Michelle. How fast did each one

drive?

Perhaps you should check these numbers in the

original statement of the problem!

Finally, let’s consider a uniform motion problem similar to those we solved in Chapter 7.

Now we have the flexibility of using two equations in two variables.

EXAMPLE 5

Larry drove 156 miles in 1 hour more than it took Mike to drive 108 miles. Mike drove at an

average rate of 2 miles per hour faster than Larry. How fast did each one travel?

444

Solution

We can represent the unknown rates and times like this:

let r represent Larry’s rate

let t represent Larry’s time

then r ϩ 2 represents Mike’s rate

and t Ϫ 1 represents Mike’s time

Because distance equals rate times time, we can set up the following system:

a

rt ϭ 156

b

(r ϩ 2)(t Ϫ 1) ϭ 108

Solving the first equation for r produces r ϭ

equation and simplifying, we obtain

156

ϩ 2b 1t Ϫ 12 ϭ 108

t

156

156 Ϫ

ϩ 2t Ϫ 2 ϭ 108

t

156

2t Ϫ

ϩ 154 ϭ 108

t

156

2t Ϫ

ϩ 46 ϭ 0

t

2t2 Ϫ 156 ϩ 46t ϭ 0

156

156

. Substituting

for r in the second

t

t

a

Multiply both sides by t, t

0

2t ϩ 46t Ϫ 156 ϭ 0

2

t2 ϩ 23t Ϫ 78 ϭ 0

We can solve this quadratic equation by factoring.

1t ϩ 2621t Ϫ 32 ϭ 0

t ϩ 26 ϭ 0

or

t ϭ -26

or

tϪ3ϭ0

tϭ3

We must disregard the negative solution. So Larry’s time is 3 hours, and Mike’s time is 3 Ϫ 1 ϭ

156

2 hours. Larry’s rate is

ϭ 52 miles per hour, and Mike’s rate is 52 ϩ 2 ϭ 54 miles per hour.

3

Problem Set 10.5

Solve each of the following problems. (Objective 1)

1. Find two consecutive whole numbers whose product

is 306.

2. Find two consecutive whole numbers whose product

is 702.

3. Suppose that the sum of two positive integers is 44 and

their product is 475. Find the integers.

4. Two positive integers differ by 6. Their product is 616.

Find the integers.

5. Find two numbers such that their sum is 6 and their

product is 4.

6. Find two numbers such that their sum is 4 and their

product is 1.

7. The sum of a number and its reciprocal is

Find the number.

322

.

2

73

8. The sum of a number and its reciprocal is . Find

24

the number.

9. Each of three consecutive even whole numbers is

squared. The three results are added and the sum

is 596. Find the numbers.

10. Each of three consecutive whole numbers is squared. The

three results are added, and the sum is 245. Find the

three whole numbers.

10.5 • Solving Problems Using Quadratic Equations

11. The sum of the square of a number and the square of

one-half of the number is 80. Find the number.

12. The difference between the square of a positive number,

and the square of one-half the number is 243. Find the

number.

13. Find the length and width of a rectangle if its length is 4

meters less than twice the width, and the area of the rectangle is 96 square meters.

14. Suppose that the length of a rectangular region is

4 centimeters greater than its width. The area of the

region is 45 square centimeters. Find the length and

width of the rectangle.

445

22. The area of a circle is numerically equal to twice the circumference of the circle. Find the length of a radius of

the circle.

23. The sum of the lengths of the two legs of a right triangle is

14 inches. If the length of the hypotenuse is 10 inches, find

the length of each leg.

24. A page for a magazine contains 70 square inches of type.

The height of a page is twice the width. If the margin

around the type is to be 2 inches uniformly, what are the

dimensions of the page?

25. A 5-by-7-inch picture is surrounded by a frame of uniform width (see Figure 10.13). The area of the picture

15. The perimeter of a rectangle is 80 centimeters, and its

area is 375 square centimeters. Find the length and width

of the rectangle.

and frame together is 80 square inches. Find the width

of the frame.

16. The perimeter of a rectangle is 132 yards and its area is

1080 square yards. Find the length and width of the rectangle.

17. The area of a tennis court is 2106 square feet (see

26

times the

9

7 inches

Figure 10.12). The length of the court is

width. Find the length and width of a tennis court.

5 inches

Figure 10.13

Figure 10.12

18. The area of a badminton court is 880 square feet. The

length of the court is 2.2 times the width. Find the length

and width of the court.

19. An auditorium in a local high school contains 300 seats.

There are 5 fewer rows than the number of seats per row.

Find the number of rows and the number of seats per

row.

20. Three hundred seventy-five trees were planted in rows

in an orchard. The number of trees per row was 10 more

than the number of rows. How many rows of trees are in

the orchard?

21. The area of a rectangular region is 63 square feet. If the

length and width are each increased by 3 feet, the area is

increased by 57 square feet. Find the length and width of

the original rectangle.

26. A rectangular piece of cardboard is 3 inches longer

than it is wide. From each corner, a square piece 2

inches on a side is cut out. The flaps are then turned up

to form an open box that has a volume of 140 cubic

inches. Find the length and width of the original piece

of cardboard.

27. A class trip was to cost \$3000. If there had been ten more

students, it would have cost each student \$25 less. How

many students took the trip?

28. Simon mowed some lawns and earned \$40. It took him 3

hours longer than he anticipated, and thus he earned \$3 per

hour less than he anticipated. How long did he expect the

mowing to take?

29. A piece of wire 56 inches long is cut into two pieces and

each piece is bent into the shape of a square. If the sum of

the areas of the two squares is 100 square inches, find the

length of each piece of wire.

30. Suppose that by increasing the speed of a car by

10 miles per hour, it is possible to make a trip of

200 miles in 1 hour less time. What was the original

speed for the trip?

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31. On a 50-mile bicycle ride, Irene averaged 4 miles per

hour faster for the first 36 miles than she did for the last

14 miles. The entire trip of 50 miles took 3 hours. Find

her rate for the first 36 miles.

Additional word problems can be found in Appendix B.

All of the problems in the Appendix marked as (10.5)

are appropriate for this section.

32. One side of a triangle is 1 foot more than twice

the length of the altitude to that side. If the area of the

triangle is 18 square feet, find the length of a side and

the length of the altitude to that side.

Thoughts Into Words

33. Return to Example 1 of this section and explain how the

problem could be solved using one variable and one equation.

34. Write a page or two on the topic “using algebra to solve

problems.”

Chapter 10 Summary

OBJECTIVE

SUMMARY

EXAMPLE

by factoring.

A quadratic equation in the variable x is

any equation that can be written in the

form ax2 ϩ bx ϩ c ϭ 0, when a, b, and c

are real numbers and a Z 0 .We can solve

using integers by factoring and applying

the property ab ϭ 0 if and only if a ϭ 0 or

b ϭ 0.

Solve x2 ϩ 4x ϭ 21 by factoring.

(Section 10.1/Objective 1)

the form x2 ϭ a.

(Section 10.1/Objective 2)

The property x2 ϭ a if and only if

x ϭ Ϯ1a can be used to solve certain

Solution

First set the equation equal to 0 and then

factor.

x2 ϩ 4x Ϫ 21 ϭ 0

(x ϩ 7)(x Ϫ 3) ϭ 0

or

xϩ7ϭ0

xϪ3ϭ0

or

x ϭ -7

xϭ3

The solution set is {Ϫ7, 3}.

Solve (2x ϩ 3)2 ϭ 15.

Solution

(2x ϩ 3)2 ϭ 15

Applying the property gives

2x ϩ 3 ϭ Ϯ 215

2x ϭ Ϫ 3 Ϯ 215

Ϫ 3 Ϯ 215

2

The solution set is

e

completing the square.

(Section 10.2/Objective 1)

You should be able to solve quadratic

equations by the method of completing the

square. To review this method, look back

over the examples in Section 10.2.

Ϫ 3 Ϫ 215 ϩ 3 ϩ 215

,

f.

2

2

Solve x2 ϩ 10x Ϫ 6 ϭ 0 by completing

the square.

Solution

First, add 6 to both sides of the equation.

x2 ϩ 10x ϭ 6

1

Now take of the coefficient of the x

2

term, 10, and square the result.

1

(10) ϭ 5

and

52 ϭ 25.

2

Add 25 to both sides of the equation.

x2 ϩ 10x ϩ 25 ϭ 6 ϩ 25

x2 ϩ 10x ϩ 25 ϭ 31

Now factor and then apply the

square-root property.

(x ϩ 5)2 ϭ 31

x ϩ 5 ϭ Ϯ231

x ϭ -5Ϯ231

The solution set is

{-5 Ϫ 231,-5 ϩ 231}.

(continued)

447

448

OBJECTIVE

SUMMARY

EXAMPLE

We usually state the quadratic formula as

Solve x2 ϩ 6x ϩ 2 ϭ 0 by the quadratic

formula.

(Section 10.3/Objective 1)

Choose the most appropriate

method for solving a quadratic equation.

(Section 10.4/Objective 1)

Ϫ b Ϯ 2b2 Ϫ 4ac

. We can use it to

2a

solve any quadratic equation that is written

in the form ax2 ϩ bx ϩ c ϭ 0. The final

answer should be reduced and have the

radical in simplest terms. Be careful when

reducing the final answer; errors are often

The three basic methods for solving

completing the square, and the quadratic

formula. Factoring only works if the

expression is factorable over the integers.

Completing the square can be very efficient

in situations where you can complete the

square without working with fractions. The

quadratic formula will work for any

Solution

For this problem, a ϭ 1, b ϭ 6, and

c ϭ 2.

Ϫ (6) Ϯ 2(6)2 Ϫ 4(1)(2)

2(1)

Ϫ 6 Ϯ 236 Ϫ 8

2

Ϫ 6 Ϯ 228

2

2(- 3 Ϯ 27)

Ϫ 6 Ϯ 227

ϭ

2

2

x ϭ -3 Ϯ 27

The solution set is {- 3 -27, -3 +27}.

Solve m2 ϩ 6m Ϫ 8 ϭ 0 using the

method that seems most appropriate.

Solution

The expression does not factor, so let’s

solve by completing the square.

m2 ϩ 6m Ϫ 8 ϭ 0

m2 ϩ 6m ϭ 8

m2 ϩ 6m ϩ 9 ϭ 8 ϩ 9

(m ϩ 3)2 ϭ 17

m ϩ 3 ϭ Ϯ 117

m ϭ -3 Ϯ117

The solution set is

{- 3 Ϫ 117, -3 ϩ 117}

Solve word problems

involving the Pythagorean

theorem and 30Њ– 60Њ right

triangles.

(Section 10.4/Objective 3)

The property x2 ϭ a if and only if

x ϭ Ϯ 2a can be used when working

with the Pythagorean theorem if the resulting equation is of the form, x2 ϭ a

Don’t forget:

1. In an isosceles right triangle, the lengths

of the two legs are equal.

2. In a 30°– 60° right triangle, the length of

the leg opposite the 30° angle is one-half

the length of the hypotenuse.

A rectangular football field measures

approximately 50 yards by 100 yards.

Find the length of a diagonal of the football field to the nearest tenth of a yard.

Solution

A diagonal of the rectangle divides the

rectangle into two right triangles. Use

the Pythagorean theorem to find the

hypotenuse of the triangle knowing that

the legs measure 50 yards and 100 yards.

a2 ϩ b2 ϭ c2

502 ϩ 1002 ϭ c2

12,500 ϭ c2

c ϭ 212500 L 111.8

The length of a diagonal to the nearest

tenth of a yard is 111.8 yards.

Chapter 10 • Review Problem Set

449

OBJECTIVE

SUMMARY

EXAMPLE

Solve word problems

Our knowledge of systems of equations

and quadratic equations provides us with a

stronger basis for solving word problems.

Find two numbers such that their sum is 8

and their product is 6.

(Section. 10.5/Objective 1)

Solution

Let n represent one number, then the

other number will be represented by

8 Ϫ n. Now we write an equation

showing the product.

n(8 Ϫ n) ϭ 6

8n Ϫ n2 ϭ 6

0 ϭ n2 Ϫ 8n ϩ 6

Using the quadratic formula we find

that the numbers are 4 Ϫ 210 and

4 ϩ 210.

Chapter 10 Review Problem Set

For Problems 1–22, solve each quadratic equation.

1. 12x ϩ 72 ϭ 25

2. x ϩ 8x ϭ -3

4. x2 ϭ 17x

5. n Ϫ

2

2

3. 21x2 Ϫ 13x ϩ 2 ϭ 0

4

ϭ -3

n

6. n2 Ϫ 26n ϩ 165 ϭ 0

7. 3a2 ϩ 7a Ϫ 1 ϭ 0

8. 4x2 Ϫ 4x ϩ 1 ϭ 0

9. 5x2 ϩ 6x ϩ 7 ϭ 0

10. 3x2 ϩ 18x ϩ 15 ϭ 0

11. 31x Ϫ 222 Ϫ 2 ϭ 4

12. x2 ϩ 4x Ϫ 14 ϭ 0

13. y2 ϭ 45

14. x1x Ϫ 62 ϭ 27

15. x2 ϭ x

}

17. n2 Ϫ 44n ϩ 480 ϭ 0

5x Ϫ 2

2

ϭ

3

xϩ1

4

5

ϩ ϭ6

21.

x

xϪ3

19.

16. n2 Ϫ 4n Ϫ 3 ϭ 6

18.

x2

ϭxϩ1

4

-1

2x ϩ 1

ϭ

3x Ϫ 1

-2

2

1

Ϫ ϭ3

22.

x

xϩ2

20.

For Problems 23–32, set up an equation or a system of

equations to help solve each problem.

23. The perimeter of a rectangle is 42 inches, and its area

is 108 square inches. Find the length and width of the

rectangle.

24. Find two consecutive whole numbers whose product

is 342.

25. Each of three consecutive odd whole numbers is

squared. The three results are added and the sum is 251.

Find the numbers.

26. The combined area of two squares is 50 square meters.

Each side of the larger square is three times as long as

a side of the smaller square. Find the lengths of the

sides of each square.

27. The difference in the lengths of the two legs of a right

triangle is 2 yards. If the length of the hypotenuse is

2113 yards, find the length of each leg.

28. Tony bought a number of shares of stock for a total of

\$720. A month later the value of the stock increased by

\$8 per share, and he sold all but 20 shares and regained

his original investment plus a profit of \$80. How many

shares did Tony sell and at what price per share?

29. A company has a rectangular parking lot 40 meters wide

and 60 meters long. They plan to increase the area of the

lot by 1100 square meters by adding a strip of equal

width to one side and one end. Find the width of the strip

30. Jay traveled 225 miles in 2 hours less time than it took

Jean to travel 336 miles. If Jay’s rate was 3 miles per hour

slower than Jean’s rate, find each rate.

31. The length of the hypotenuse of an isosceles right triangle is 12 inches. Find the length of each leg.

32. In a 30Њ– 60Њ right triangle, the side opposite the 60Њ angle

is 8 centimeters long. Find the length of the hypotenuse.

For more practice with word problems, consult

Appendix B. All Appendix problems that have a Chapter

10 reference would be appropriate for you to work on.

Chapter 10 Test

1. The two legs of a right triangle are 4 inches and 6 inches

long. Find the length of the hypotenuse. Express your

15. n1n Ϫ 282 ϭ - 195

2. A diagonal of a rectangular plot of ground measures 14

meters. If the width of the rectangle is 5 meters, find the

length to the nearest meter.

17. 12x ϩ 1213x Ϫ 22 ϭ -2

3. A diagonal of a square piece of paper measures

10 inches. Find, to the nearest inch, the length of a side

of the square.

19. 14x Ϫ 122 ϭ 27

16. n ϩ

3

19

ϭ

n

4

18. 17x ϩ 222 Ϫ 4 ϭ 21

20. n2 Ϫ 5n ϩ 7 ϭ 0

4. In a 30Њ– 60Њ right triangle, the side opposite the 30Њ

angle is 4 centimeters long. Find the length of the side

For Problems 5–20, solve each equation.

5. 13x ϩ 222 ϭ 49

6. 4x2 ϭ 64

For Problems 21– 25, set up an equation or a system of

equations to help solve each problem.

21. A room contains 120 seats. The number of seats per row

is 1 less than twice the number of rows. Find the number

of seats per row.

22. Abu rode his bicycle 56 miles in 2 hours less time than

it took Stan to ride his bicycle 72 miles. If Abu’s rate

was 2 miles per hour faster than Stan’s rate, find Abu’s

rate.

7. 8x2 Ϫ 10x ϩ 3 ϭ 0

8. x2 Ϫ 3x Ϫ 5 ϭ 0

9. n2 ϩ 2n ϭ 9

10. 12x Ϫ 122 ϭ -16

23. Find two consecutive odd whole numbers whose product is 255.

24. The combined area of two squares is 97 square feet.

Each side of the larger square is 1 foot more than twice

the length of a side of the smaller square. Find the

length of a side of the larger square.

11. y2 ϩ 10y ϭ 24

12. 2x2 Ϫ 3x Ϫ 4 ϭ 0

13.

4

xϪ2

ϭ

3

xϩ1

14.

2

1

5

ϩ ϭ

x

xϪ1

2

450

25. Dee bought a number of shares of stock for a total of

\$160. Two weeks later, the value of the stock had

increased \$2 per share, and she sold all but 4 shares and

regained her initial investment of \$160. How many

11

11.1 Equations and

Inequalities Involving

Absolute Value

11.2 3 ϫ 3 Systems of

Equations

11.3 Fractional Exponents

11.4 Complex Numbers

Complex Solutions

11.6 Pie, Bar, and Line

Graphs

11.7 Relations and

Functions

11.8 Applications of

Functions

Be sure that you understand the

meaning of the markings on both

the horizontal and vertical axes.

We include this chapter to give you the opportunity to expand your knowledge of topics presented in earlier chapters. From the list of section titles, the

topics may appear disconnected; however, each section is a continuation of a

topic presented in a previous chapter.

Section 11.1 continues the development of techniques for solving equations

and inequalities, which was the focus of Chapter 3. Section 11.2 uses the method

of elimination by addition from Section 8.6 to solve systems containing three

linear equations in three variables. Section 11.3 is an extension of the work we

did with exponents in Section 5.6 and with radicals in Chapter 9. Sections 11.4

and 11.5 enhance the study of quadratic equations from Chapter 10. Sections

11.6 – 11.8 extend our work with coordinate geometry from Chapter 8.

Video tutorials based on section learning objectives are available in a variety of

delivery modes.

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