Equations with Radicals

A radical equation has the variable inside a root: , , . The idea for solving is simple — undo the root by raising both sides to the matching power — but square roots come with a catch that no earlier lesson has: squaring both sides can create solutions that do not satisfy the original equation.

Those false answers are called extraneous solutions, and the check that catches them is mandatory here, not optional. This is the lesson Linear Equations warned you about when it said checking would eventually become required.

The Radical Equation Formula

Squaring both sides removes the square root. The resulting equation is often linear or quadratic. Because squaring can introduce extraneous solutions, each answer must be tested in the original equation.

The Method: One Square Root

1. Isolate the radical on one side by itself.
2. Square both sides. ; the other side is squared as a whole.
3. Solve the resulting equation.
4. Check every candidate in the original equation. Discard any that fail.

Two Square Roots: Square Twice

1. Isolate one radical, square both sides. One radical remains.
2. Isolate the remaining radical, square again.
3. Solve, then check every candidate.

Higher Roots

For a cube root, cube both sides: . Odd roots are reversible over the reals, so a single cube-root equation does not produce extraneous solutions — though checking is still wise.

Why Squaring Creates False Solutions

The equation carries a hidden requirement: the principal square root on the left is never negative, so must be for any solution to be real. Squaring both sides throws that requirement away. After squaring, is satisfied equally by the case and by the case , because both produce the same square. Any candidate that actually belonged to the second case is extraneous.

That gives you a way to anticipate trouble. In Worked Example B, needs ; the candidate violates that before you even substitute, so it is doomed. Checking the sign of the isolated non-radical side against each candidate is often faster than a full substitution, though the full substitution is the definitive test.

Spotting a “No Solution” Early

If isolating the radical leaves , stop — there is no real solution, and squaring would only manufacture a false one. Likewise, if the radicand itself can never be non‑negative for any real (rare, but it happens with expressions like ), the solution set is empty. Recognizing these at a glance saves a page of algebra.

Worked Example A: One Square Root, Linear Result

Solve .

Step 1 — the radical is already isolated. Square both sides:

Step 2 — solve:

Step 3 — check: . ✓

Answer: .

Worked Example B: An Extraneous Solution Appears

Solve .

Step 1 — square both sides:

Step 2 — standard form and factor:

Step 3 — check both in the original:

• : . ✓
• : , but the right side is . . ✗ extraneous

Answer: only. The value solves the squared equation but not the original.

Worked Example C: Two Square Roots

Solve .

Step 1 — isolate one radical:

Step 2 — square both sides:

Step 3 — square again:

Step 4 — check: . ✓

Answer: .

Worked Example D: A Cube Root

Solve .

Step 1 — cube both sides:

Step 2 — solve:

Step 3 — check: . ✓

Answer: . No extraneous solution risk with an odd root.

Worked Example E: Both Candidates Fail

Solve .

Step 1 — isolate the radical:

The isolated square root equals a negative number. A principal square root is never negative, so there is no real solution — the answer is .

If you had missed that and squared anyway, you would get , so . Checking: . The lone candidate is extraneous, confirming the empty solution set. This is why the “isolated radical equals a negative” check comes before squaring.

Common Mistakes to Avoid

• Skipping the check. With square roots, checking is required. An unchecked extraneous solution is a wrong answer.
• Squaring before isolating. squared as-is gives , which still has a radical. Isolate first.
• Squaring term by term. , not .
• Forgetting the middle term. Squaring a binomial with a radical always produces a cross term: .
• Accepting a negative isolated radical. has no real solution; do not square and report an answer.
• Missing that a “no real solution” answer is complete. An empty solution set is a legitimate final answer.

Where This Shows Up Later

• Equations Reducible to Quadratic Form. is quadratic in ; the extraneous-solution logic carries over.
• Distance and rate formulas. and similar relationships are solved by squaring.
• Distance formula and circle equations. Setting two distance expressions equal produces a radical equation.
• Functions and domains. A radical function’s domain is exactly the set where the radicand is nonnegative — the same condition that makes a candidate solution valid.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. Solve .

Problem 13. Solve .

Problem 14. Solve .

Quick Reference

The root-and-power rules behind this are in Radicals. When a square root sits inside a quadratic-shaped equation, see Equations Reducible to Quadratic Form. The squared-out equations are finished with factoring or the quadratic formula. More Algebra lessons are available too.