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Math Practice

Radical Equations Practice Problems

Twelve hand-built radical equations with full worked solutions. Each isolates the radical, squares (or cubes) both sides, solves, and — for square roots — checks every candidate in the original equation.

  1. 1.

    Solve .

    Show full solution

    Check: . ✓

  2. 2.

    Solve .

    Show full solution

    Check: . ✓

  3. 3.

    Solve .

    Show full solution

    Isolate: . Square: . Check: . ✓

  4. 4.

    Solve .

    Show full solution

    A principal square root is never negative. No real solution.

  5. 5.

    Solve .

    Show full solution

    Check: gives . ✓ gives . ✗

    Answer: .

  6. 6.

    Solve .

    Show full solution

    Square: . Check: . ✓

  7. 7.

    Solve .

    Show full solution

    Check: and . ✓

  8. 8.

    Solve .

    Show full solution

    Cube: . Check: . ✓

  9. 9.

    Solve .

    Show full solution

    Isolate: . Square: , so , . Square: . Check: . ✓

  10. 10.

    Solve .

    Show full solution

    Check: gives . ✓ gives . ✗

    Answer: .

  11. 11.

    Solve .

    Show full solution

    Isolate: . Square: , so , .

    Check: gives and . ✓ makes . ✗

    Answer: .

  12. 12.

    Solve .

    Show full solution

    Square: , so , . Check: and . ✓
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Radical Equations Practice Problems

The Equations with Radicals lesson covers one radical, two radicals, higher roots, and the mandatory check; this set is 12 equations to practice on, each fully solved.

How to use this set

Isolate the radical, raise both sides to the matching power, solve, and — for square roots — check every candidate in the original equation. Enter the surviving solution and press Check Answers. If every candidate turns out to be extraneous, type no solution — that is a complete answer here, not a blank.

Common mistakes to watch for

Skipping the check on square-root equations.

Squaring before isolating. squared as-is still has a radical.

Squaring term by term. .

Squaring an isolated radical that equals a negative and reporting the result anyway.

Where to go next

For radical problems that don’t involve an equation, such as simplifying, multiplying and rationalizing, use the Radicals Practice Problems. Equations that are quadratic in are the Equations Reducible to Quadratic Form Practice Problems. The squared-out equations finish with factoring or the Quadratic Formula Practice Problems. For the full method, see the Equations with Radicals lesson.

Frequently Asked Questions

How do I solve an equation with a square root?+

Isolate the radical, square both sides, solve the result, then substitute every answer back into the original equation to discard extraneous solutions.

Why do extraneous solutions appear?+

Squaring is not reversible: implies , but so does . Squaring can satisfy as , which fails the original. Checking catches these.

What if there are two square roots?+

Isolate one radical, square; isolate the other, square again. Two radicals usually means squaring twice with an ordinary equation in between.

Do cube-root equations need an extraneous check?+

Cubing is reversible for real numbers, so a single cube-root equation doesn't introduce extraneous solutions. Checking is still good practice.

Can a radical equation have no solution?+

Yes. If the isolated radical equals a negative number, like , there is no real solution. Sometimes every candidate turns out extraneous.

What do I type when every candidate is extraneous?+

Type no solution. That is the complete, correct answer for a radical equation whose candidates all fail the check — ∅ and none are accepted too. Leaving the box empty is not the same thing and will be marked wrong.

More practice problems

Equations Reducible to Quadratic Form Practice Problems Quadratic Formula Practice Problems Simplifying Radicals Practice Problems