# Radical Equations

An equation in which the unknown appears in a radicand is called a **radical equation**.

For instance: and are radical equations.

To solve a radical equation, begin by isolating the most complicated radical expression on one side of the equation, and then eliminate the radical by raising both sides of the equation to a power equal to the index of the radical. You may have to repeat this technique in order to eliminate all radicals. When the equation is radical-free, simplify and solve it. Since extraneous roots may be introduced when both sides of an equation are raised to even powers, all roots must be checked by substitution in the original equation whenever a radical with an even index is involved.

__Example__:

Solve .

__Solution__:

To isolate the radical, we add to both sides of the equation

Now we raise both sides to the power and obtain

or

It follows that . Since we did not raise both sides of the equation to an even power, it is not necessary to check our solution, but it is good practice to do so anyway. Substituting in the left side of the original equation, we obtain

Hence, the solution is correct.

__Example__:

Solve .

__Solution__:

We add to both sides of the equation to isolate on the left side. Thus

Now we square both sides of the equation to obtain

The equation still contains a radical, so we simplify and isolate this radical

Again, we square both sides, so we get

or

**To Check:**

Hence, is a solution.

**To Check:**

Therefore, is an extraneous root that was introduced by squaring both sides of the equation. The only solution