Method guideCalculus I · Algebra IIIntermediate

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.

LaTeX article Updated July 13, 2026

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

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 example

Common mistakes

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

Keep these ideas

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