Solve Radical Equations

Solving Radical Equations

Radical equations are equations where the variable appears inside a root, such as a square root. The main idea is to isolate the radical first, then remove it by raising both sides to the appropriate power.

Method

1. Isolate the radical term on one side of the equation if possible.
2. Remove the root by squaring both sides for square roots. If a higher root appears, use the matching power.
3. Solve the resulting equation. This may give one or more candidate solutions.
4. Check every answer in the original equation. Squaring can create extraneous solutions, so verification is essential.

Important check

A solution must make both sides of the original equation equal and must keep any expression under an even root nonnegative. If a proposed value causes a negative radicand in a square root, it cannot be a valid solution.

Example strategy

If you have an equation like a radical plus a number equals another expression, first get the radical alone. Then square both sides carefully, simplify, and solve. Afterward, substitute each solution back into the original equation.

Final step

Always report only the values that survive the check and simplify the answer completely.