How To Solve Radical Equations

By The Calcumatix Team Reviewed by Calcumatix Editorial Review 3 min read Updated July 23, 2026

Quick Answer

To solve a radical equation, isolate the radical term on one side of the equation, raise both sides to the power that matches the root (square for a square root, cube for a cube root), then solve the resulting equation normally. Because raising both sides to a power can introduce false answers, every candidate solution must be substituted back into the original equation to confirm it actually works before being accepted as final.

A radical equation contains the variable inside a square root, cube root, or other root symbol. Solving it requires eliminating the radical before applying standard algebra, but that elimination step can introduce false answers along the way. This guide walks through the three-step procedure, isolate, raise to a power, solve and check, and shows a real example of the extraneous-root problem that makes the final check step essential, not optional. To skip the manual math, use the radical equation calculator to get the result instantly.

Step by step:

Isolate the radical term on one side of the equation.
Raise both sides to the power that matches the root index.
Solve the resulting equation, then substitute every candidate answer back into the original equation to check for extraneous solutions.

Step One: Isolate The Radical On One Side

Before removing the root, rearrange the equation so the radical term sits completely alone on one side. Add or subtract any other terms so only the radical expression remains isolated, moving everything else to the opposite side of the equals sign. This setup ensures the next step eliminates the radical cleanly.

Step Two: Raise Both Sides To The Matching Power

With the radical isolated, apply the exponent that matches the root index to both sides of the equation. Squaring both sides eliminates a square root; cubing both sides eliminates a cube root. This step converts the radical equation into an ordinary algebraic equation.

Step Three: Solve The Resulting Equation And Check Every Answer

Solve the new equation using standard algebra to isolate the variable. Depending on the equation, this may produce one answer or, if a quadratic results, two candidate answers. Substitute every candidate back into the ORIGINAL radical equation, not the squared version, since only the original equation reveals whether an answer is genuinely valid or extraneous.

Worked Example: A Straightforward Square Root Equation

Worked example: Original equation: √(x + 3) - 2 = 5. Step 1 (isolate the radical): Add 2 to both sides: √(x + 3) = 7. Step 2 (square both sides): (√(x + 3))² = 7² becomes x + 3 = 49. Step 3 (solve and check): Subtract 3 from both sides: x = 46. Checking: √(46 + 3) - 2 = √49 - 2 = 7 - 2 = 5, which matches the original equation.

Final answer: x = 46. The check confirms this is a genuine solution, not extraneous.

Worked Example: An Equation With An Extraneous Solution

Not every candidate survives the check step. This example shows why skipping it is a real risk, not a formality.

Worked example: Original equation: √(x + 2) = x - 4. Step 1 (already isolated): The radical is already alone on the left side. Step 2 (square both sides): (√(x + 2))² = (x - 4)² becomes x + 2 = x² - 8x + 16. Step 3 (solve the quadratic): Rearranging gives x² - 9x + 14 = 0, which factors to (x - 2)(x - 7) = 0, producing two candidates: x = 2 and x = 7.

Checking both candidates in the original equation: For x = 7: √(7 + 2) = 7 - 4 becomes √9 = 3, which is 3 = 3, true. For x = 2: √(2 + 2) = 2 - 4 becomes √4 = -2, which is 2 = -2, false.

Final answer: x = 7 only. x = 2 is an extraneous solution, a byproduct of squaring that solves the squared equation but fails the original. It must be discarded.

What Causes An Extraneous Solution?

Squaring both sides of an equation can turn a false statement into a true one, because squaring erases the distinction between a positive and a negative number. In the second example above, squaring -2 and 2 both produce 4, which is why the algebra could not detect that -2 does not equal the principal (nonnegative) square root of 4. This is precisely why the check step against the original, unsquared equation is required on every radical equation, not just as good practice. See the math calculators hub for related tools.

Sources and References

Disclaimer: This guide is for educational and informational purposes only. It provides general mathematical procedures and should not replace formal academic instruction. Consult your textbook or mathematics instructor for specific academic requirements.

Frequently asked questions

How do you solve an equation with a cube root?

Follow the same three steps used for a square root: isolate the radical, then raise both sides to the power matching the root index. For a cube root, cube both sides instead of squaring them, since cubing is the operation that undoes a cube root.

Can a radical equation have two valid solutions?

Yes. Squaring both sides of an equation sometimes produces a quadratic equation, which can yield two candidate solutions. Both must be substituted back into the original radical equation separately, since one may check out while the other turns out to be extraneous.

What happens if a square root equals a negative number?

A real square root is never negative, since the principal square root function only returns nonnegative values. If isolating the radical produces an equation where the square root equals a negative number, the equation has no real solution, and squaring both sides anyway would only generate an extraneous answer.

Why must the radical be isolated before squaring?

Squaring an equation that still has other terms on the same side as the radical introduces extra cross terms and often leaves a radical in the result. Isolating the radical first ensures that squaring eliminates it cleanly in one step, rather than complicating the equation further.

How do you solve an equation with two radical terms?

Isolate one radical on each side of the equation, then square both sides to eliminate at least one of them. The result usually still contains one radical term, so the isolate-and-square process is typically repeated a second time to eliminate it as well.