Solving Radical Equations Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

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

A "radical" equation is an equation in which the variable is hiding inside

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.

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.

Square both sides.

Graphical check:

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

2.

Square both sides.

Graphical check:

Graphs do not intersect.
No solution.

Let's see an example where BOTH answers check:

3.

Square both sides.

Graphical check:

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

4.

Square both sides.

Graphical check:

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

5.

Square both sides.

Graphical check:

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

6.

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

Graphical check:

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

7.

Square both sides.