Radical Equations

Isolate the radical, square both sides, and check for extraneous roots.

The idea

A radical equation contains the unknown under a square root. Squaring removes the root, since $(\sqrt{u})^{2} = u$ for $u \ge 0$, so the method is: isolate the radical on one side, square both sides, and solve the ordinary equation that results.

Squaring, however, discards information. Two different numbers can share a square — $2$ and $-2$ both square to $4$ — so the squared equation may accept values the original does not. Such a value is called extraneous. It arises because the symbol $\sqrt{u}$ means the principal square root, which is never negative: an equation asking $\sqrt{u}$ to equal a negative number has no solution, but squaring erases the sign, and the squared equation accepts the value anyway.

The final step is therefore part of the method. Substitute every candidate into the original equation and keep only those that satisfy it.

Ways to work on it

• Walkthrough. Square both sides to solve three radical equations.
• Practice. Solve a single radical equation by squaring.
• Hardest. Solve an equation with an extraneous root and identify it.

Not sure where to start? Take the ten-question placement test.