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Elementary Algebra 2e

10.1 Solve Quadratic Equations Using the Square Root Property

Elementary Algebra 2e10.1 Solve Quadratic Equations Using the Square Root Property
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 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
  13. 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 10.1

Before you get started, take this readiness quiz.

Simplify: 7575.
If you missed this problem, review Example 9.12.

Be Prepared 10.2

Simplify: 643643.
If you missed this problem, review Example 9.67.

Be Prepared 10.3

Factor: 4x212x+94x212x+9.
If you missed this problem, review Example 7.43.

Quadratic equations are equations of the form ax2+bx+c=0ax2+bx+c=0, where a0a0. They differ from linear equations by including a term with the variable raised to the second power. 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 use three other methods to solve quadratic equations.

Solve Quadratic Equations of the Form ax2 = 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=9x2=9.

x2=9Put the equation in standard form.x29=0Factor the left side.(x3)(x+3)=0Use the Zero Product Property.(x3)=0,(x+3)=0Solve each equation.x=3,x=−3Combine the two solutions into±form.x=±3(The solution is readxis equal to positive or negative 3.’)x2=9Put the equation in standard form.x29=0Factor the left side.(x3)(x+3)=0Use the Zero Product Property.(x3)=0,(x+3)=0Solve each equation.x=3,x=−3Combine the two solutions into±form.x=±3(The solution is readxis equal to positive or negative 3.’)

We can easily use factoring to find the solutions of similar equations, like x2=16x2=16 and x2=25x2=25, because 16 and 25 are perfect squares. But what happens when we have an equation like x2=7x2=7? Since 7 is not a perfect square, we cannot solve the equation by factoring.

These equations are all of the form x2=kx2=k.
We defined the square root of a number in this way:

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

This leads to the Square Root Property.

Square Root Property

If x2=kx2=k, and k0k0, then x=korx=kx=korx=k.

Notice that the Square Root Property gives two solutions to an equation of the form x2=kx2=k: the principal square root of kk and its opposite. We could also write the solution as x=±kx=±k.

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

x2=9Use the Square Root Property.x=±9Simplify the radical.x=±3Rewrite to show the two solutions.x=3,x=−3x2=9Use the Square Root Property.x=±9Simplify the radical.x=±3Rewrite to show the two solutions.x=3,x=−3

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

Use the Square Root Property.x2=7x=±7Rewrite to show two solutions.x=7,x=7We cannot simplify7,so we leave the answer as a radical.Use the Square Root Property.x2=7x=±7Rewrite to show two solutions.x=7,x=7We cannot simplify7,so we leave the answer as a radical.

Example 10.1

Solve: x2=169x2=169.

Try It 10.1

Solve: x2=81x2=81.

Try It 10.2

Solve: y2=121y2=121.

Example 10.2

How to Solve a Quadratic Equation of the Form ax2=kax2=k Using the Square Root Property

Solve: x248=0x248=0.

Try It 10.3

Solve: x250=0x250=0.

Try It 10.4

Solve: y227=0y227=0.

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.

To use the Square Root Property, the coefficient of the variable term must equal 1. In the next example, we must divide both sides of the equation by 5 before using the Square Root Property.

Example 10.3

Solve: 5m2=805m2=80.

Try It 10.5

Solve: 2x2=982x2=98.

Try It 10.6

Solve: 3z2=1083z2=108.

The Square Root Property started by stating, ‘If x2=kx2=k, and k0k0’. What will happen if k<0k<0? This will be the case in the next example.

Example 10.4

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

Try It 10.7

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

Try It 10.8

Solve: d2+81=0d2+81=0.

Remember, we first isolate the quadratic term and then make the coefficient equal to one.

Example 10.5

Solve: 23u2+5=1723u2+5=17.

Try It 10.9

Solve: 12x2+4=2412x2+4=24.

Try It 10.10

Solve: 34y23=1834y23=18.

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

Example 10.6

Solve: 2c24=452c24=45.

Try It 10.11

Solve: 5r22=345r22=34.

Try It 10.12

Solve: 3t2+6=703t2+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 like (x3)2=16(x3)2=16, too. We will treat the whole binomial, (x3)(x3), as the quadratic term.

Example 10.7

Solve: (x3)2=16(x3)2=16.

Try It 10.13

Solve: (q+5)2=1(q+5)2=1.

Try It 10.14

Solve: (r3)2=25(r3)2=25.

Example 10.8

Solve: (y7)2=12(y7)2=12.

Try It 10.15

Solve: (a3)2=18(a3)2=18.

Try It 10.16

Solve: (b+2)2=40(b+2)2=40.

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

Example 10.9

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

Try It 10.17

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

Try It 10.18

Solve: (y34)2=716.(y34)2=716.

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

Example 10.10

Solve: (x2)2+3=30(x2)2+3=30.

Try It 10.19

Solve: (a5)2+4=24(a5)2+4=24.

Try It 10.20

Solve: (b3)28=24(b3)28=24.

Example 10.11

Solve: (3v7)2=−12(3v7)2=−12.

Try It 10.21

Solve: (3r+4)2=−8(3r+4)2=−8.

Try It 10.22

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)2a(xh)2. But they are perfect square trinomials, so we will factor to put them in the form we need.

Example 10.12

Solve: p210p+25=18p210p+25=18.

Try It 10.23

Solve: x26x+9=12x26x+9=12.

Try It 10.24

Solve: y2+12y+36=32y2+12y+36=32.

Example 10.13

Solve: 4n2+4n+1=164n2+4n+1=16.

Try It 10.25

Solve: 9m212m+4=259m212m+4=25.

Try It 10.26

Solve: 16n2+40n+25=416n2+40n+25=4.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with solving quadratic equations:

Section 10.1 Exercises

Practice Makes Perfect

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

In the following exercises, solve the following quadratic equations.

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.

x2+20=0x2+20=0

10.

y2+64=0y2+64=0

11.

25a2+3=1125a2+3=11

12.

32b27=4132b27=41

13.

7p2+10=267p2+10=26

14.

2q2+5=302q2+5=30

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

In the following exercises, solve the following quadratic equations.

15.

(x+2)2=9(x+2)2=9

16.

(y5)2=36(y5)2=36

17.

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

18.

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

19.

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

20.

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

21.

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

22.

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

23.

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

24.

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

25.

(5c+1)2=−27(5c+1)2=−27

26.

(8d6)2=−24(8d6)2=−24

27.

m24m+4=8m24m+4=8

28.

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

29.

25x230x+9=3625x230x+9=36

30.

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

Mixed Practice

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

31.

2r2=322r2=32

32.

4t2=164t2=16

33.

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

34.

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

35.

9w224w+16=19w224w+16=1

36.

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

37.

a218=0a218=0

38.

b2108=0b2108=0

39.

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

40.

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

41.

m2+12=0m2+12=0

42.

n2+48=0n2+48=0

43.

u214u+49=72u214u+49=72

44.

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

45.

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

46.

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

47.

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

48.

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

49.

6c2+4=296c2+4=29

50.

2d24=772d24=77

51.

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

52.

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

Everyday Math

53.

Paola has enough mulch to cover 48 square feet. She wants to use it to make three square vegetable gardens of equal sizes. Solve the equation 3s2=483s2=48 to find ss, the length of each garden side.

54.

Kathy is drawing up the blueprints for a house she is designing. She wants to have four square windows of equal size in the living room, with a total area of 64 square feet. Solve the equation 4s2=644s2=64 to find ss, the length of the sides of the windows.

Writing Exercises

55.

Explain why the equation x2+12=8x2+12=8 has no solution.

56.

Explain why the equation y2+8=12y2+8=12 has two solutions.

Self Check

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

This table has three rows and four columns. The first row is a header row and it labels each column. The first column is labeled “I can …”, the second “Confidently”, the third “With some help” and the last “No–I don’t get it”. In the “I can…” column the next row reads “solve quadratic equations of the form a x squared equals k using the square root property.” and the last row reads “solve quadratic equations of the form a times the quantity x minus h squared equals k using the square root property.” The remaining columns are blank.

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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