Solve radical equations and check for extraneous solutions. | Nonlinear equations in one variable | SAT Math

Radical equations need squaring and a final check for extraneous solutions.

Core Idea

Isolate the radical, square both sides to eliminate it, solve the resulting equation, then check every answer in the original equation. Squaring can create solutions that don't actually work — these are extraneous solutions.

Understanding

Workflow: isolate first, square second, check last.

If the radical is not isolated, move everything else away from it.
Square both sides to remove the radical.
Solve the resulting equation.
Plug every candidate back into the original equation and reject any that fail.

If a radical equation has two radicals, isolate one radical, square, then isolate again if needed. The check at the end is what protects you from extraneous solutions.

Step by Step

Isolate the radical on one side of the equation.
Square both sides to remove the radical.
Solve the resulting equation (often linear or quadratic).
Substitute each solution into the original equation to verify — discard any that fail.

Misconceptions

Assuming all solutions from the squared equation are valid. Extraneous solutions appear frequently on the SAT.
Squaring before isolating the radical. If $\sqrt 𝑥 + 3 + 1 = 4$ , you must subtract 1 first to get $\sqrt 𝑥 + 3 = 3$ , then square.
Forgetting that $\sqrt 𝑥$ by convention refers to the non-negative root only. $\sqrt 9 = 3$ , not $\pm 3$ .

Question

Worked Example

What is the solution set of $\sqrt 3 𝑥 + 1 = 𝑥 - 1$ ?

Select an answer to see the explanation