(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
TI83+/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:
Let's see an example where the answer does
NOT check:
Let's see an example where BOTH answers
check:
Let's see an example where ONLY ONE answer
checks:
Let's see an example where there are
radicals on both sides:
Let's see an example where the radicals are
multiplied by constants:
Let's see an example involving x^{2}
in the problem:
