Radical Equations and Extraneous Solutions

Algebra Foundation

Algebra 2

Explore advanced functions, complex numbers, polynomials, rational and radical expressions, exponentials, logarithms, conics, sequences, introductory trigonometry, probability, and statistics.

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Algebra 2 · Unit 5 · Topic 5.4

Radical Equations and Extraneous Solutions

Isolate a radical before raising both sides to a power.

Definition and explanation

Isolate a radical before raising both sides to a power. Squaring can introduce extra solutions, so enforce the original domain and substitute all candidates into the original equation.

Worked example

Solve √(x + 6) = x. The right side must be nonnegative. Squaring gives x + 6 = x², hence (x − 3)(x + 2) = 0. Reject −2 because √4 ≠ −2; x = 3 works because √9 = 3.

Common mistake

Squaring is not automatically reversible when the sides may have different signs.

Key takeaway

Isolate a radical before raising both sides to a power.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.