Method guideAlgebra IIIntermediate11 min read

Solving radical equations and checking for extraneous roots

Isolate one radical, raise both sides to the matching power, solve, and check every candidate in the original equation.

\sqrt{f(x)}=g(x)\Rightarrow f(x)=g(x)^2

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Updated July 13, 2026

WarningStart here

Squaring is not a reversible step over all real expressions, so it can introduce answers the original equation never had.

Isolate before raising powers

Move other terms away so one radical stands alone. Squaring a sum that contains a radical creates a cross term and more algebra than necessary.

If multiple radicals remain, isolate and square in stages rather than trying to eliminate everything at once.

\sqrt{x+4}-2=x\Rightarrow\sqrt{x+4}=x+2

Domain conditions predict trouble

An even-index radicand must be nonnegative, and an isolated principal square root is also nonnegative. Therefore the other side must be nonnegative too.

These conditions can eliminate candidates early, but the original-equation check is still required.

f(x)\ge0,\quad g(x)\ge0

Why extraneous roots appear

If a = b then a² = b², but the reverse allows a = −b as well. Squaring loses sign information.

Substituting into the original equation restores that information and separates genuine solutions from artifacts.

a^2=b^2\Rightarrow a=\pm b

Worked exampleSquare, solve, verify

Solve √(x + 6) = x.

x\ge0

The isolated square root is nonnegative.

x+6=x^2

Square both sides.

x^2-x-6=0=(x-3)(x+2)

Move all terms and factor.

x=3\ \text{or}\ x=-2

List algebraic candidates.

\sqrt9=3\quad\text{but}\quad\sqrt4\ne-2

Check in the original equation.

Result

\boxed{x=3}

Watch for

Common mistakes

Squaring before isolating the radical.
Forgetting domain conditions.
Keeping every root of the squared equation.

Keep

Three takeaways

Isolate first.
Even roots impose nonnegative conditions.
Check every candidate in the original equation.