Solving Radical Equations

A radical equation has a variable trapped inside a square root (or other radical). Solving one means squaring both sides to remove the radical — but squaring can introduce solutions that don't actually work in the original equation, called extraneous solutions. That's why the last step is always a check.

1. Isolate the radical on one side of the equation.
2. Square both sides to eliminate the radical.
3. Solve the resulting equation (often a linear or quadratic equation).
4. Check every solution in the ORIGINAL equation — not the squared version.

Solve √(2x + 3) = x. The radical is already isolated. Squaring both sides gives 2x + 3 = x², which rearranges to x² − 2x − 3 = 0, factoring to (x−3)(x+1) = 0, giving x = 3 or x = −1.

Checking in the original equation: for x = 3, √9 = 3, which is true — keep it. For x = −1, √1 = 1, but 1 ≠ −1 — this solution is extraneous and must be rejected. The only valid solution is x = 3.

Radical equations often build on earlier radical skills, like multiplying and dividing radicals to simplify a term before isolating it, or recognizing the shape of a square root function in the equation.

Squaring both sides of an equation is not a reversible step: it can turn a false statement (like 1 = −1) into a true one (1² = (−1)²). That's the mathematical root of the extraneous solution — and it's why radical functions and their equations always require a final check.

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• Algebra