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Radical Equations

A free Algebra II lesson from the “Rational Exponents and Radicals” unit, with a worked example and practice problems including step-by-step solutions.

To solve a radical equation, isolate the radical, square both sides, then solve. Squaring can introduce extraneous solutions, so check answers in the original equation.

What you'll learn

Why it matters: Pipe-flow formulas, pendulum-period equations, and engineering safety factors need a squared step to remove a radical. Always check candidate answers against the original equation — squaring can invent solutions that do not actually work.

Worked example

Problem. Solve sqrt(x + 5) = 4.

  1. Square both sides.
  2. x + 5 = 16.
  3. x = 11, and sqrt(16) = 4 checks.

Answer: 11

Practice problems

1. Solve sqrt(x) = 9.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve sqrt(x) = 9.
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. Square both sides.
  4. Check the result by substituting or estimating: the response should match 81 and make sense in the original problem.

Answer: 81

2. Solve sqrt(x + 1) = 5.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve sqrt(x + 1) = 5.
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. x + 1 = 25.
  4. Check the result by substituting or estimating: the response should match 24 and make sense in the original problem.

Answer: 24

3. Solve sqrt(x - 3) = 6.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve sqrt(x - 3) = 6.
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. x - 3 = 36.
  4. Check the result by substituting or estimating: the response should match 39 and make sense in the original problem.

Answer: 39

4. Solve sqrt(2x) = 8.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve sqrt(2x) = 8.
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. 2x = 64.
  4. Check the result by substituting or estimating: the response should match 32 and make sense in the original problem.

Answer: 32

5. Why check radical equation solutions?

Choices: Squaring can create extraneous solutions · Square roots never have answers · All answers are negative · Checking is optional only for graphs

Show solution
  1. Challenge: First identify exactly what the question is asking: Why check radical equation solutions?
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. Squaring is not always reversible.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Squaring can create extraneous solutions

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