Concept 6

Solve equations involving radicals; check for extraneous solutions.

Isolate the radical, square carefully, and check for extraneous solutions.

Core Idea

Isolate the radical before squaring, and always check the result in the original equation because squaring can create an answer that was never valid.

Understanding

Rule: Radical equations feel routine until the final check. Once you square both sides, you may have introduced a false solution.

On ACT, the right workflow is short: isolate the radical, square carefully, solve the resulting equation, then substitute each candidate back into the original radical equation. Do not skip that last step.

Step by Step

  1. Isolate the radical.
  2. Square both sides and solve the resulting equation.
  3. Check every candidate in the original equation.

Misconceptions

  • Assuming every solution from the squared equation is valid.
  • Squaring before isolating the radical.
  • Skipping the final check.
Question

Worked Example

What is the solution to √𝑥+9 =𝑥 −3?

Select an answer to see the explanation