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Intermediate Algebra 2e

9.4 Solve Quadratic Equations in Quadratic Form

Intermediate Algebra 2e9.4 Solve Quadratic Equations in Quadratic Form
  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 equations in quadratic form
Be Prepared 9.10

Before you get started, take this readiness quiz.

Factor by substitution: y4y220.y4y220.
If you missed this problem, review Example 6.21.

Be Prepared 9.11

Factor by substitution: (y4)2+8(y4)+15.(y4)2+8(y4)+15.
If you missed this problem, review Example 6.22.

Be Prepared 9.12

Simplify: x12·x14x12·x14 (x13)2(x13)2 (x−1)2.(x−1)2.
If you missed this problem, review Example 8.33.

Solve Equations in Quadratic Form

Sometimes when we factored trinomials, the trinomial did not appear to be in the ax2 + bx + c form. So we factored by substitution allowing us to make it fit the ax2 + bx + c form. We used the standard uu for the substitution.

To factor the expression x4 − 4x2 − 5, we noticed the variable part of the middle term is x2 and its square, x4, is the variable part of the first term. (We know (x2)2=x4.(x2)2=x4.) So we let u = x2 and factored.

.
.
Let u=x2u=x2 and substitute. .
Factor the trinomial. .
Replace u with x2x2. .

Similarly, sometimes an equation is not in the ax2 + bx + c = 0 form but looks much like a quadratic equation. Then, we can often make a thoughtful substitution that will allow us to make it fit the ax2 + bx + c = 0 form. If we can make it fit the form, we can then use all of our methods to solve quadratic equations.

Notice that in the quadratic equation ax2 + bx + c = 0, the middle term has a variable, x, and its square, x2, is the variable part of the first term. Look for this relationship as you try to find a substitution.

Again, we will use the standard u to make a substitution that will put the equation in quadratic form. If the substitution gives us an equation of the form ax2 + bx + c = 0, we say the original equation was of quadratic form.

The next example shows the steps for solving an equation in quadratic form.

Example 9.30

How to Solve Equations in Quadratic Form

Solve: 6x47x2+2=06x47x2+2=0

Try It 9.59

Solve: x46x2+8=0x46x2+8=0.

Try It 9.60

Solve: x411x2+28=0x411x2+28=0.

We summarize the steps to solve an equation in quadratic form.

How To

Solve equations in quadratic form.

  1. Step 1. Identify a substitution that will put the equation in quadratic form.
  2. Step 2. Rewrite the equation with the substitution to put it in quadratic form.
  3. Step 3. Solve the quadratic equation for u.
  4. Step 4. Substitute the original variable back into the results, using the substitution.
  5. Step 5. Solve for the original variable.
  6. Step 6. Check the solutions.

In the next example, the binomial in the middle term, (x − 2) is squared in the first term. If we let u = x − 2 and substitute, our trinomial will be in ax2 + bx + c form.

Example 9.31

Solve: (x2)2+7(x2)+12=0.(x2)2+7(x2)+12=0.

Try It 9.61

Solve: (x5)2+6(x5)+8=0.(x5)2+6(x5)+8=0.

Try It 9.62

Solve: (y4)2+8(y4)+15=0.(y4)2+8(y4)+15=0.

In the next example, we notice that (x)2=x.(x)2=x. Also, remember that when we square both sides of an equation, we may introduce extraneous roots. Be sure to check your answers!

Example 9.32

Solve: x3x+2=0.x3x+2=0.

Try It 9.63

Solve: x7x+12=0.x7x+12=0.

Try It 9.64

Solve: x6x+8=0.x6x+8=0.

Substitutions for rational exponents can also help us solve an equation in quadratic form. Think of the properties of exponents as you begin the next example.

Example 9.33

Solve: x232x1324=0.x232x1324=0.

Try It 9.65

Solve: x235x1314=0.x235x1314=0.

Try It 9.66

Solve: x128x14+15=0.x128x14+15=0.

In the next example, we need to keep in mind the definition of a negative exponent as well as the properties of exponents.

Example 9.34

Solve: 3x−27x−1+2=0.3x−27x−1+2=0.

Try It 9.67

Solve: 8x−210x−1+3=0.8x−210x−1+3=0.

Try It 9.68

Solve: 6x−223x−1+20=0.6x−223x−1+20=0.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with solving quadratic equations.

Section 9.4 Exercises

Practice Makes Perfect

Solve Equations in Quadratic Form

In the following exercises, solve.

155.

x47x2+12=0x47x2+12=0

156.

x49x2+18=0x49x2+18=0

157.

x413x230=0x413x230=0

158.

x4+5x236=0x4+5x236=0

159.

2x45x2+3=02x45x2+3=0

160.

4x45x2+1=04x45x2+1=0

161.

2x47x2+3=02x47x2+3=0

162.

3x414x2+8=03x414x2+8=0

163.

(x3)25(x3)36=0(x3)25(x3)36=0

164.

(x+2)23(x+2)54=0(x+2)23(x+2)54=0

165.

(3y+2)2+(3y+2)6=0(3y+2)2+(3y+2)6=0

166.

(5y1)2+3(5y1)28=0(5y1)2+3(5y1)28=0

167.

(x2+1)25(x2+1)+4=0(x2+1)25(x2+1)+4=0

168.

(x24)24(x24)+3=0(x24)24(x24)+3=0

169.

2(x25)25(x25)+2=02(x25)25(x25)+2=0

170.

2(x25)27(x25)+6=02(x25)27(x25)+6=0

171.

xx20=0xx20=0

172.

x8x+15=0x8x+15=0

173.

x+6x16=0x+6x16=0

174.

x+4x21=0x+4x21=0

175.

6x+x2=06x+x2=0

176.

6x+x1=06x+x1=0

177.

10x17x+3=010x17x+3=0

178.

12x+5x3=012x+5x3=0

179.

x23+9x13+8=0x23+9x13+8=0

180.

x233x13=28x233x13=28

181.

x23+4x13=12x23+4x13=12

182.

x2311x13+30=0x2311x13+30=0

183.

6x23x13=126x23x13=12

184.

3x2310x13=83x2310x13=8

185.

8x2343x13+15=08x2343x13+15=0

186.

20x2323x13+6=020x2323x13+6=0

187.

x8x12+7=0x8x12+7=0

188.

2x7x12=152x7x12=15

189.

6x−2+13x−1+5=06x−2+13x−1+5=0

190.

15x−226x−1+8=015x−226x−1+8=0

191.

8x−22x−13=08x−22x−13=0

192.

15x−24x−14=015x−24x−14=0

Writing Exercises

193.

Explain how to recognize an equation in quadratic form.

194.

Explain the procedure for solving an equation in quadratic form.

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 equations in quadratic form.” “Confidently,” “with some help,” or “No, I don’t get it.”

On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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