Lesson 4

Lesson 4: Solving Radical Equations

 

Focus

 

Crash investigators can use the length of skid marks to determine the speed of a vehicle when the vehicle first applied its brakes. Investigators use radical equations, such as and , where s is speed, d is skid distance, f is drag factor, and v is velocity (or speed). This is just one of the many practical applications of radical equations.

 

In this lesson you will learn to solve equations containing radicals. The first three lessons in this module have prepared you to master this essential mathematical skill.

 

Outcomes

 

At the end of this lesson you will be able to

• determine any restrictions on values for the variable in a radical equation

• determine the roots of a radical equation algebraically, and explain the process used to solve the equation

• verify by substitution that the values determined in solving a radical equation algebraically are roots of the equation

• explain why some roots determined in solving a radical equation algebraically are extraneous

Lesson Questions

 

You will investigate the following questions:

• How do you determine the roots of a radical equation algebraically?
  
  
• How do you verify that the values determined in solving a radical equation are viable roots of the equation?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 4 Assignment (Download the Lesson 4 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Try This and Share activities
  
  
• additions to Module 5 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.

Before beginning this lesson you should be able to

• solve equations for a given variable
  
  
• factor radical binomials and trinomials
  
  
• use the quadratic formula to solve a quadratic equation

Are You Ready?

 

Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Rearrange the following equations to isolate and solve for the variable.
   
   
   1. 5x + 1 = 7 + 3x Answer
      
      
   2. 4(x − 2) = 3(2x − 7) Answer
      
      
   3. Answer
      
      
   4. Answer
      
      
2. Factor the following expressions.
   
   
   1. Answer
      
      
   2. 4x2 − 9y2 Answer
      
      
   3. a2 − 2a + 3 Answer
      
      
   4. 3n2 − 2n − 5 Answer
      
      
   5. 16x2 − Answer
      
      
3. Find the roots of the following equations using the quadratic formula.
   
   
   1. x2 + 4x − 21 = 0 Answer
      
      
   2. 3x2 + 6x = −10 Answer
      
      
   3. −3x2 + 12x + 1 = 0 Answer

How did the questions go? If you feel comfortable with the concepts covered in the questions, skip forward to Discover. If you experienced difficulties, use the resources in Refresher to review these important concepts before continuing through the lesson.

Refresher

 

To gain confidence manipulating equations to isolate a specific variable, watch "Rearrange formulas to isolate specific variables."

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Factoring radical binomials and trinomials uses the same techniques as factoring other binomials and trinomials. Factoring Special Products shows the techniques you can use.

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Watch “Using the Quadratic Formula” to review how to use the quadratic formula.

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Work through The Quadratic Formula interactive lesson.

Go back to the Are You Ready section and try the questions again. If you are still having difficulty, contact your teacher.

 

Discover

 

In this section you will

• solve an equation with one radical
  
  
• solve an equation with two radicals

Try This 1

1.  
   1. Solve the radical equation 7 − = 3.
      
      
   2. Verify your solution by substituting the answer into the original equation to see if the expression on the left side is equal to the expression on the right side.
      
      
2.  
   1. Solve the radical equation .
      
      
   2. Verify your solution by substituting the answer into the original equation to see if the expression on the left side is equal to the expression on the right side. Use a calculator if you wish.
      
      
3. What is different between solving these equations and solving a regular linear equation?

 Save your work in your course folder.

 

Share 1

 

Based on your observations from Try This 1, respond to the following questions with a partner or group.

1. When the equation contains a variable inside the radical sign, what do you need to do differently to solve the equation?
   
   
2. Summarize your discussion by creating a general rule about solving equations where the variable is under a radical sign.

 Save your work in your course folder.

Try This 2

 

Open Relations and Functions: Radical Equations. Work through the introduction, tutorials, and example questions. Think about the following questions as you do:

• How do the steps used to solve the radical equations (finding the roots) in the applet compare to the steps you used in Try This 1?
  
  
• What are extraneous roots and how are they determined?
  
  
• What techniques can help you solve radical equations containing two radicals?

Example 1: Equations that Include a Radical Term

Example 3: Solving Radical Equations with the Quadratic Equation

 

Solving radical equations can also include the need to use the quadratic equation. For example, how would you find the roots of the following equation?

 

 

First you need to notice that x2 − 3 > 0. When solutions are found they need to be checked against this requirement.

 

 

a = 1, b = −3, c = −3

 

The roots of the quadratic equation are  and . However since  is less than 3, it cannot be a root of the original equation.  is greater than 3 and is the only permissible root of the original equation.

 

Example 4: Equations with Two Radicals

Self-Check 2

1. What restrictions are there on the values of the variables in  if the equation involves real numbers? Answer
   
   
2. Solve
   
   
   1. by isolating  Answer
      
      
   2. by isolating  Answer
      
      
   3. Check each for extraneous roots. Answer
      
      
3. In question 2, which method was easier when solving for d? Explain. Answer
   
   
4. Solve  and check for extraneous roots Answer

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Skip forward to Connect if you feel you have a solid understanding of how to

• determine any restrictions on values for the variable in a radical equation
  
  
• determine the roots of a radical equation algebraically, and explain the process used to solve the equation
  
  
• verify by substitution that the values determined in solving a radical equation algebraically are roots of the equation
  
  
• explain why some roots determined in solving a radical equation algebraically are extraneous
  

If you need a bit more practice, complete Self-Check 3.

 

Self-Check 3

Connect

 

Lesson 4 Assignment