# How do extraneous solutions arise from radical equations?

##### 1 Answer

In general, extraneous solutions arise when we perform non-invertible operations on both sides of an equation. (That is, they sometimes arise, but not always.)

Squaring (or raising to any other even power) is a non-invertible operation. Solving equations involving square roots involves squaring both sides of an equation.

**Example 1** : To show the idea:

The equations:

Square both sides of

Square both sides of

which is equivalent to

and, rewriting the left,

So the solution set is

This time, it is

**Example2** : Extraneous solution.

Solve

Subtracting

Squaring (!) gives

This requires,

Factoring to get

finds the solution set to be

Checking these reveals that

**Example 3** : No extraneous solution.

Solve

Squaring (!) gives,

Which leads to