Radical & Rational Exponents

Solving Radical Equations

Solving equations containing a radical expression, checking for extraneous solutions.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Rational Exponent Form

When Squaring Both Sides Can Lie to You

Consider the equation x = −3. It's true for exactly one value: x = −3. Before reading on, square both sides and solve the resulting equation, x² = 9 — does it still have exactly one solution, or has something changed?

x² = 9 has two solutions, x = 3 and x = −3, even though the original equation, x = −3, was only ever true for one of them. Squaring both sides isn't a fully reversible move — it can't tell the difference between a number and its opposite, since both square to the same result. That's why solving a radical equation by squaring both sides can introduce an extraneous solution: a value that satisfies the squared equation but not the original one.

Definition — Extraneous Solution

A value produced by solving an equation that doesn't actually satisfy the original equation — typically introduced by a step, like squaring, that isn't fully reversible. Checking every solution in the original equation is the only way to catch one.

Worked Example — Solving a Radical Equation

Solve √(x + 6) = x. Square both sides: x + 6 = x². Rearrange: x² − x − 6 = 0. Factor: (x − 3)(x + 2) = 0, giving x = 3 or x = −2. Check both in the original equation: √(3 + 6) = √9 = 3, matching x = 3 — valid. √(−2 + 6) = √4 = 2, but x = −2 — these don't match, since 2 ≠ −2. x = −2 is extraneous and must be rejected. The only true solution is x = 3.

Worked Example — Understanding Why the Extraneous Solution Appeared

In the equation above, why did x = −2 pass the squared equation but fail the original? Squaring √(x + 6) = x turned it into (√(x + 6))² = x², which is also satisfied whenever √(x + 6) = −x — a different equation than the original. x = −2 happens to satisfy that second, unintended equation, which is exactly the kind of extra solution squaring can smuggle in.

Tip

Never skip the final check when solving a radical equation — it's not optional bookkeeping, since a genuinely valid-looking algebraic solution can still fail to satisfy the original equation.

Common Mistakes

Reporting every solution found after squaring, without checking each one against the original equation.
Always substitute every candidate solution back into the original (unsquared) equation — an extraneous solution will satisfy the squared version but fail the original.
Squaring only one side of the equation, or squaring a sum incorrectly, such as treating (x + 6) under a radical as squaring to x + 36 instead of (√(x+6))² = x + 6 directly.
Squaring √(x + 6) simply removes the radical, leaving x + 6 exactly — there's no additional squaring of the terms inside, since the square root and the square directly undo each other.

Key Takeaways

Squaring both sides of an equation isn't fully reversible, since it can't distinguish a number from its opposite.
This can introduce extraneous solutions — values satisfying the squared equation but not the original.
Every solution to a radical equation must be checked against the original equation before being accepted.

Summary

Extraneous solutions are a genuine hazard whenever an equation-solving step isn't fully reversible — a theme that will reappear with logarithmic and rational equations later in this course. The next unit turns to exponential functions, where quantities grow or shrink by a constant multiplying factor.

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