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Intermediate Algebra 2e

9.1 Solve Quadratic Equations Using the Square Root Property

Intermediate Algebra 2e9.1 Solve Quadratic Equations Using the Square Root Property
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Solve quadratic equations of the form ax2=kax2=k using the Square Root Property
  • Solve quadratic equations of the form a(xh)2=ka(xh)2=k using the Square Root Property
Be Prepared 9.1

Before you get started, take this readiness quiz.

Simplify: 128.128.
If you missed this problem, review Example 8.13.

Be Prepared 9.2

Simplify: 325325.
If you missed this problem, review Example 8.50.

Be Prepared 9.3

Factor: 9x212x+49x212x+4.
If you missed this problem, review Example 6.23.

A quadratic equation is an equation of the form ax2 + bx + c = 0, where a0a0. Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form ax2. We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.

We have seen that some quadratic equations can be solved by factoring. In this chapter, we will learn three other methods to use in case a quadratic equation cannot be factored.

Solve Quadratic Equations of the form ax2=kax2=k using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation x2 = 9.

x2=9x2=9
Put the equation in standard form. x29=0x29=0
Factor the difference of squares. (x3)(x+3)=0(x3)(x+3)=0
Use the Zero Product Property. x3=0x3=0x3=0x3=0
Solve each equation. x=3x=−3x=3x=−3

We can easily use factoring to find the solutions of similar equations, like x2 = 16 and x2 = 25, because 16 and 25 are perfect squares. In each case, we would get two solutions, x=4,x=−4x=4,x=−4 and x=5,x=−5.x=5,x=−5.

But what happens when we have an equation like x2 = 7? Since 7 is not a perfect square, we cannot solve the equation by factoring.

Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. Also, (−13)2 = 169, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169. So, every positive number has two square roots—one positive and one negative. We earlier defined the square root of a number in this way:

Ifn2=m,thennis a square root ofm.Ifn2=m,thennis a square root ofm.

Since these equations are all of the form x2 = k, the square root definition tells us the solutions are the two square roots of k. This leads to the Square Root Property.

Square Root Property

If x2 = k, then

x=korx=korx=±k.x=korx=korx=±k.

Notice that the Square Root Property gives two solutions to an equation of the form x2 = k, the principal square root of kk and its opposite. We could also write the solution as x=±k.x=±k. We read this as x equals positive or negative the square root of k.

Now we will solve the equation x2 = 9 again, this time using the Square Root Property.

x2=9x2=9
Use the Square Root Property. x=±9x=±9
x=±3x=±3
Sox=3orx=−3.Sox=3orx=−3.

What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation x2 = 7.

x2=7x2=7
Use the Square Root Property. x=7,x=7x=7,x=7

We cannot simplify 77, so we leave the answer as a radical.

Example 9.1 How to solve a Quadratic Equation of the form ax2 = k Using the Square Root Property

Solve: x250=0.x250=0.

Try It 9.1

Solve: x248=0.x248=0.

Try It 9.2

Solve: y227=0.y227=0.

The steps to take to use the Square Root Property to solve a quadratic equation are listed here.

How To

Solve a quadratic equation using the square root property.

  1. Step 1. Isolate the quadratic term and make its coefficient one.
  2. Step 2. Use Square Root Property.
  3. Step 3. Simplify the radical.
  4. Step 4. Check the solutions.

In order to use the Square Root Property, the coefficient of the variable term must equal one. In the next example, we must divide both sides of the equation by the coefficient 3 before using the Square Root Property.

Example 9.2

Solve: 3z2=108.3z2=108.

Try It 9.3

Solve: 2x2=98.2x2=98.

Try It 9.4

Solve: 5m2=80.5m2=80.

The Square Root Property states ‘If x2=kx2=k,’ What will happen if k<0?k<0? This will be the case in the next example.

Example 9.3

Solve: x2+72=0x2+72=0.

Try It 9.5

Solve: c2+12=0.c2+12=0.

Try It 9.6

Solve: q2+24=0.q2+24=0.

Our method also works when fractions occur in the equation, we solve as any equation with fractions. In the next example, we first isolate the quadratic term, and then make the coefficient equal to one.

Example 9.4

Solve: 23u2+5=17.23u2+5=17.

Try It 9.7

Solve: 12x2+4=24.12x2+4=24.

Try It 9.8

Solve: 34y23=18.34y23=18.

The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator.

Example 9.5

Solve: 2x28=41.2x28=41.

Try It 9.9

Solve: 5r22=34.5r22=34.

Try It 9.10

Solve: 3t2+6=70.3t2+6=70.

Solve Quadratic Equations of the Form a(xh)2 = k Using the Square Root Property

We can use the Square Root Property to solve an equation of the form a(xh)2 = k as well. Notice that the quadratic term, x, in the original form ax2 = k is replaced with (xh).

On the left is the equation a times x square equals k. Replacing x in this equation with the expression x minus h changes the equation. It is now a times the square of x minus h equals k.

The first step, like before, is to isolate the term that has the variable squared. In this case, a binomial is being squared. Once the binomial is isolated, by dividing each side by the coefficient of a, then the Square Root Property can be used on (xh)2.

Example 9.6

Solve: 4(y7)2=48.4(y7)2=48.

Try It 9.11

Solve: 3(a3)2=54.3(a3)2=54.

Try It 9.12

Solve: 2(b+2)2=80.2(b+2)2=80.

Remember when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.

Example 9.7

Solve: (x13)2=59.(x13)2=59.

Try It 9.13

Solve: (x12)2=54.(x12)2=54.

Try It 9.14

Solve: (y+34)2=716.(y+34)2=716.

We will start the solution to the next example by isolating the binomial term.

Example 9.8

Solve: 2(x2)2+3=57.2(x2)2+3=57.

Try It 9.15

Solve: 5(a5)2+4=104.5(a5)2+4=104.

Try It 9.16

Solve: 3(b+3)28=88.3(b+3)28=88.

Sometimes the solutions are complex numbers.

Example 9.9

Solve: (2x3)2=−12.(2x3)2=−12.

Try It 9.17

Solve: (3r+4)2=−8.(3r+4)2=−8.

Try It 9.18

Solve: (2t8)2=−10.(2t8)2=−10.

The left sides of the equations in the next two examples do not seem to be of the form a(xh)2. But they are perfect square trinomials, so we will factor to put them in the form we need.

Example 9.10

Solve: 4n2+4n+1=16.4n2+4n+1=16.

Try It 9.19

Solve: 9m212m+4=25.9m212m+4=25.

Try It 9.20

Solve: 16n2+40n+25=4.16n2+40n+25=4.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with using the Square Root Property to solve quadratic equations.

Section 9.1 Exercises

Practice Makes Perfect

Solve Quadratic Equations of the Form ax2 = k Using the Square Root Property

In the following exercises, solve each equation.

1.

a2=49a2=49

2.

b2=144b2=144

3.

r224=0r224=0

4.

t275=0t275=0

5.

u2300=0u2300=0

6.

v280=0v280=0

7.

4m2=364m2=36

8.

3n2=483n2=48

9.

43x2=4843x2=48

10.

53y2=6053y2=60

11.

x2+25=0x2+25=0

12.

y2+64=0y2+64=0

13.

x2+63=0x2+63=0

14.

y2+45=0y2+45=0

15.

43x2+2=11043x2+2=110

16.

23y28=−223y28=−2

17.

25a2+3=1125a2+3=11

18.

32b27=4132b27=41

19.

7p2+10=267p2+10=26

20.

2q2+5=302q2+5=30

21.

5y27=255y27=25

22.

3x28=463x28=46

Solve Quadratic Equations of the Form a(xh)2 = k Using the Square Root Property

In the following exercises, solve each equation.

23.

(u6)2=64(u6)2=64

24.

(v+10)2=121(v+10)2=121

25.

(m6)2=20(m6)2=20

26.

(n+5)2=32(n+5)2=32

27.

(r12)2=34(r12)2=34

28.

(x+15)2=725(x+15)2=725

29.

(y+23)2=881(y+23)2=881

30.

(t56)2=1125(t56)2=1125

31.

(a7)2+5=55(a7)2+5=55

32.

(b1)29=39(b1)29=39

33.

4(x+3)25=274(x+3)25=27

34.

5(x+3)27=685(x+3)27=68

35.

(5c+1)2=−27(5c+1)2=−27

36.

(8d6)2=−24(8d6)2=−24

37.

(4x3)2+11=−17(4x3)2+11=−17

38.

(2y+1)25=−23(2y+1)25=−23

39.

m24m+4=8m24m+4=8

40.

n2+8n+16=27n2+8n+16=27

41.

x26x+9=12x26x+9=12

42.

y2+12y+36=32y2+12y+36=32

43.

25x230x+9=3625x230x+9=36

44.

9y2+12y+4=99y2+12y+4=9

45.

36x224x+4=8136x224x+4=81

46.

64x2+144x+81=2564x2+144x+81=25

Mixed Practice

In the following exercises, solve using the Square Root Property.

47.

2r2=322r2=32

48.

4t2=164t2=16

49.

(a4)2=28(a4)2=28

50.

(b+7)2=8(b+7)2=8

51.

9w224w+16=19w224w+16=1

52.

4z2+4z+1=494z2+4z+1=49

53.

a218=0a218=0

54.

b2108=0b2108=0

55.

(p13)2=79(p13)2=79

56.

(q35)2=34(q35)2=34

57.

m2+12=0m2+12=0

58.

n2+48=0.n2+48=0.

59.

u214u+49=72u214u+49=72

60.

v2+18v+81=50v2+18v+81=50

61.

(m4)2+3=15(m4)2+3=15

62.

(n7)28=64(n7)28=64

63.

(x+5)2=4(x+5)2=4

64.

(y4)2=64(y4)2=64

65.

6c2+4=296c2+4=29

66.

2d24=772d24=77

67.

(x6)2+7=3(x6)2+7=3

68.

(y4)2+10=9(y4)2+10=9

Writing Exercises

69.

In your own words, explain the Square Root Property.

70.

In your own words, explain how to use the Square Root Property to solve the quadratic equation (x+2)2=16(x+2)2=16.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can solve quadratic equations of the form a times x squared equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”

Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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