(This lesson will work primarily with radicals that are square roots.)

A "radical" equation is an equation in which the variable is hiding inside
a radical symbol (in the radicand).


is a radical equation


is NOT a radical equation
 


 

To solve radical equations:

  
1.  Isolate the radical (or one of the radicals) to one side of the
     equal sign.
2.  If the radical is a square root, square each side of the
     equation. (If the radical is not a square root, raise each side to a power
       equal to the index of the root.)
3.  Solve the resulting equation.
4.  Check your answer(s) to avoid extraneous roots.

How to use your
TI-83+/84+ graphing calculator
 with radical equations.
Click here.



When working with radical equations (that are square roots),  ...

1. you must square sides, not terms.   Consider:
 

          

BUT ...

                

       
2. you must check your answers.  The process of squaring the sides of an equation creates a "derived" equation which may not be equivalent to the original radical equation.  Consequently, solving this new derived equation may create solutions that never previously existed.  These "extra" roots that are not true solutions of the original radical equation are called extraneous roots and are rejected as answers.  Consider:
 

The first statement is false, but when each side is squared, the concluding statement is true.

 
 

The first statement is false, but when each side is squared, the concluding statement is true.


 You can "see" this problem of extraneous roots by graphing.  Consider
There is NO solution to this equation since the graphs do not intersect.  Squaring both sides, however, creates two graphs that DO intersect which leads to a false answer.
                                               
   
                                 Graph of original equation components.             Graph of squares.

 

Examples:
Let's see an example where the answer checks:

1.        
Answer:  x = 5
 

Square both sides.

(Answer checks!)

 Graphical check:

Let's see an example where the answer does NOT check:

2.      
Answer:  no solution
 

Isolate the radical.
Square both sides.

(Answer does NOT check!)

 Graphical check:

    Graphs do not intersect. 
          No solution.

Let's see an example where BOTH answers check:

3.
Answer:  x = 5, 2
 

Square both sides.

      
(BOTH Answers check!)

 Graphical check:

Let's see an example where ONLY ONE answer checks:

4.     
Answer:  x = 7
 

Square both sides.

   

(Only ONE Answer checks!)

 Graphical check:

Let's see an example where there are radicals on both sides:

5.     
Answer:  x = 2
 

Square both sides.

(Answer checks!)

 Graphical check:

Let's see an example where the radicals are multiplied by constants:

6.
Answer:  x = 10
 

Square both sides.
Be careful of those numbers in front.

(Answer checks!)

 Graphical check:

Let's see an example involving x2 in the problem:

7.     
Answer:  x = 5, -4
 

Square both sides.

         
(BOTH Answers check!)

 Graphical check: