An extraneous solution is a solution that appears to be valid when solving an equation, but it actually does not satisfy the original equation or problem. Extraneous solutions often occur when solving equations involving square roots, logarithms, or other inverse functions.
To check for extraneous solutions, one can substitute the potential solution back into the original equation and see if it satisfies the equation. If the substitute yields a true statement, then the solution is valid. If it yields a false statement, then the solution is extraneous.
For example, consider the equation √(x + 4) = 3. Solving for x, one might obtain x = 5. However, this solution is extraneous because x + 4 must be non-negative, and 5 + 4 = 9 is positive. Substituting x = 5 back into the original equation, we get √(5 + 4) = √9, which is not equal to 3. Thus, x = 5 is an extraneous solution.