Radical Equation
A radical equation contains a variable inside a radical (such as a square root), solved by isolating the radical and raising both sides to the appropriate power.
Definition
A radical equation has a variable inside a radical; solving it means isolating the radical, raising both sides to the power matching its index, solving the resulting equation, and then checking every solution in the original equation, since raising both sides to a power is not a reversible (equivalence) step and can introduce false, extraneous solutions. Formally, isolating a radical of the form $f(x)^{1/n} = g(x)$ and raising both sides to the $n$-th power gives $f(x) = g(x)^n$, a many-to-one operation when $n$ is even, so the true solution set is the intersection of the solutions to $f(x) = g(x)^n$ with the domain constraint $g(x) \ge 0$ needed to match the principal-root convention.
Example
$\sqrt{x + 3} = 5$: squaring gives $x + 3 = 25$, so $x = 22$; checking, $\sqrt{22+3} = 5$, correct. $\sqrt{2x-1} = x - 2$: squaring gives $x^2 - 6x + 5 = 0 = (x-1)(x-5)$, so $x = 1$ or $x = 5$, but checking $x = 1$ gives $\sqrt{1}=1$ against $1-2=-1$ (no match, extraneous), while $x = 5$ gives $\sqrt{9}=3$ against $5-2=3$ (correct). $\sqrt{x} = -3$ has no real solution at all, even though squaring gives $x = 9$, since the principal square root $\sqrt{9} = 3$ can never equal $-3$.
Key Insight
Always check solutions back in the original equation, squaring both sides can manufacture "fake" answers called extraneous solutions, because squaring throws away sign information. This many-to-one behavior of even powers, and the multi-valued nature of n-th roots more generally (every non-zero complex number has $n$ distinct n-th roots), is handled rigorously in complex analysis through Riemann surfaces.