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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 \(\sqrt{x + 5} = 4\).

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    $$ x + 5 = 16 \quad\Longrightarrow\quad x = 11 $$

    Check: \(\sqrt{16} = 4\). ✓

  2. 2.

    Solve \(\sqrt{3x - 2} = 5\).

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    $$ 3x - 2 = 25 \quad\Longrightarrow\quad x = 9 $$

    Check: \(\sqrt{25} = 5\). ✓

  3. 3.

    Solve \(\sqrt{x} + 3 = 8\).

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    Isolate: \(\sqrt{x} = 5\). Square: \(x = 25\). Check: \(5 + 3 = 8\). ✓

  4. 4.

    Solve \(\sqrt{x - 2} = -3\).

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

  5. 5.

    Solve \(\sqrt{x + 6} = x\).

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    $$ x + 6 = x^{2} \quad\Longrightarrow\quad x^{2} - x - 6 = 0 \quad\Longrightarrow\quad (x - 3)(x + 2) = 0 $$

    Check: \(x = 3\) gives \(\sqrt{9} = 3\). ✓ \(x = -2\) gives \(\sqrt{4} = 2 \neq -2\). ✗

    Answer: \(x = 3\).

  6. 6.

    Solve \(\sqrt{2x + 3} = \sqrt{x + 10}\).

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    Square: \(2x + 3 = x + 10 \Rightarrow x = 7\). Check: \(\sqrt{17} = \sqrt{17}\). ✓

  7. 7.

    Solve \(\sqrt{x^{2} + 5} = x + 1\).

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    $$ x^{2} + 5 = x^{2} + 2x + 1 \quad\Longrightarrow\quad 4 = 2x \quad\Longrightarrow\quad x = 2 $$

    Check: \(\sqrt{9} = 3\) and \(2 + 1 = 3\). ✓

  8. 8.

    Solve \(\sqrt[3]{2x + 1} = 3\).

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    Cube: \(2x + 1 = 27 \Rightarrow x = 13\). Check: \(\sqrt[3]{27} = 3\). ✓

  9. 9.

    Solve \(\sqrt{x + 9} - \sqrt{x} = 1\).

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    Isolate: \(\sqrt{x + 9} = 1 + \sqrt{x}\). Square: \(x + 9 = 1 + 2\sqrt{x} + x\), so \(8 = 2\sqrt{x}\), \(\sqrt{x} = 4\). Square: \(x = 16\). Check: \(\sqrt{25} - \sqrt{16} = 5 - 4 = 1\). ✓

  10. 10.

    Solve \(x = \sqrt{x + 20}\).

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    $$ x^{2} = x + 20 \quad\Longrightarrow\quad x^{2} - x - 20 = 0 \quad\Longrightarrow\quad (x - 5)(x + 4) = 0 $$

    Check: \(x = 5\) gives \(\sqrt{25} = 5\). ✓ \(x = -4\) gives \(\sqrt{16} = 4 \neq -4\). ✗

    Answer: \(x = 5\).

  11. 11.

    Solve \(\sqrt{2x + 5} + 1 = x\).

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    Isolate: \(\sqrt{2x + 5} = x - 1\). Square: \(2x + 5 = x^{2} - 2x + 1\), so \(x^{2} - 4x - 4 = 0\), \(x = 2 \pm 2\sqrt{2}\).

    Check: \(x = 2 + 2\sqrt{2} \approx 4.83\) gives \(\sqrt{2x + 5} \approx \sqrt{14.66} \approx 3.83\) and \(x - 1 \approx 3.83\). ✓ \(x = 2 - 2\sqrt{2} \approx -0.83\) makes \(x - 1 < 0\). ✗

    Answer: \(x = 2 + 2\sqrt{2}\).

  12. 12.

    Solve \(\sqrt{4x + 1} = 2\sqrt{x} + 1\).

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    Square: \(4x + 1 = 4x + 4\sqrt{x} + 1\), so \(0 = 4\sqrt{x}\), \(x = 0\). Check: \(\sqrt{1} = 1\) and \(0 + 1 = 1\). ✓
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Mathovia Team
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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

These problems are hand-written, not auto-generated. Isolate the radical, raise both sides to the matching power, solve, and — for square roots — check every candidate in the original equation. Then compare with the solution.

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

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.

Why is this a curated set?+

Answers can be irrational, and the extraneous check makes single-number auto-checking unreliable. Each problem has a full solution instead.

More practice problems

Equations Reducible to Quadratic Form Practice Problems Quadratic Formula Practice Problems Simplifying Radicals Practice Problems: Free Practice Extracting Perfect Powers