TABLE OF CONTENTS

1.4  Radical Equations and Functions

Prerequisite knowledge and skills:

        a working knowledge and understanding of simplifying radicals

Terms to know:

·         Cube root

·         Equivalent equations

·         Extraneous solution

·         Index

·         Perfect cubes                                 

·         Perfect squares                                     

·         Quadratic equation

·         Radical

·         Radical sign

·         Radicand

·         Rational number

·         Square root

·         Term  

Skill prep:  Solving quadratic equations;  Radicals to rational exponents; Rational exponent to radicals

Concept prep:  Solving simple radical equations

The following are examples of radical equations.

The equation

is NOT a radical equation, even though it contains radicals because the variable is NOT under the radical sign.

How did you solve #3a - 3j in the concept prep exercises ?

One way is to raise the entire equation (both sides) to an exponent, thus eliminating the radical sign.    Remember, exponents and radicals are inverse operations, meaning one undoes the other.  To eliminate a square root, square the equation; to eliminate a cube root, cube the equation, etc.

Eliminating radicals:  More detail

When the equation is raised to an exponent, each SIDE must be raised to the exponent NOT each term individually.

Common Errors

Example 1.  Solve     

Square both sides:                                                           

Solve for x:                                                                    

Check:                                                                                       

Check by graphing:

Note that if we use our graphing calculator and set , we obtain a graph like the following:  (The graph below was created with a software product called MAPLE.  Your calculator graph may not have as good resolution.)

.

Notice that the point of intersection occurs at .  What does this tell us about the solution?

Example 2.  Solve   

Isolate the radical term:                                                                 

Square both sides:                                                              

Solve for x:                                                                                

Check:                                                                                                 

Check by graphing:

On your graphing calculator, set  and your viewing window to be [-10,50] by [-10, 20] to obtain a graph like the following.

Note that the point of intersection has an x-coordinate of 38.  What does this tell us about our algebraic solution?

More detail

SOLVING RADICAL EQUATIONS CONTAINING TWO RADICALS

The following example has two radicals and is thus more involved.  The basic procedure is the same, though we need to square both sides of the equation twice.

We still need to isolate (either) one of the radicals first. 

Example 3.  Solve   

Isolate one radical term:                                                

Square both sides:                                                    

Notice that the right side is a binomial,

so we need to be careful to square it correctly,

using binomial multiplication, or FOIL:                                                                                                                        Combine  like terms here

                                                                         So,     

Combine like terms:                                                 

Now we’re going to repeat the entire procedure by isolating the radical, then squaring both sides again.

Isolate the radical that’s left by                

subtracting 14 and m from both sides :                                                                   

Square both sides again:                                                   

                                                                            We don’t need to FOIL here, just use rules of exponents.

Get one side = 0:                                                                                 

                                                                                                         (3.1)

To make factoring easier, we divide (3.1) through by 9:   

Factor:                                                                        

Set each factor to 0 and solve:                                      

Check in the original equation:

   For   m = 11:                           

So m = 11 is NOT a solution.  This happened because we squared both sides of the equation.

 For   m = -1:                        

So m= -1 is the only solution.      

Note that the following graph with  supports our algebraic solution:

                                                                                                Y1

                                       Y2

More Worked Examples

Activity with graphs

Homework

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