← Chapter 8
College Algebra · Chapter 8

When Squaring Invents Answers

To free a variable from under a radical you square both sides — there is no gentler way. But squaring is a one-way door: if a = b then a² = b², yet a² = b² is also true when a = −b. So the squared equation quietly answers a second question you never asked, and its answers come back mixed in with yours. Those imposters are called extraneous solutions. Step through each problem one line at a time and watch the graph show you exactly where the phantom comes from.

The picture

Substitute back into the original

candidate—
left side—
right side—
A candidate is only a solution if it survives this test.
Radical equationspick a problem, then step through it

The method, every time

1. Isolate one radical alone on one side.  2. Square both sides — the whole side, expanded, not term by term.  3. Solve whatever is left; if a radical remains, isolate and square again.  4. Check every candidate in the original equation and throw out the ones that fail. Step 4 is not optional politeness — it is part of solving.