Radical Equation
AlgebraA 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 square root (or other root). To solve it, isolate the radical, then square both sides to remove it. Always check your answer.
Example
$\sqrt{x + 3} = 5$. Square both sides: $x + 3 = 25$, so $x = 22$. Check: $\sqrt{22 + 3} = \sqrt{25} = 5$. Correct.
Key Insight
Always check your solutions back in the original equation. Squaring both sides can introduce "fake" answers called extraneous solutions.
Definition
A radical equation contains a variable under a radical sign. Solution steps: (1) isolate the radical on one side; (2) raise both sides to the power equal to the index of the radical; (3) solve the resulting equation; (4) check all solutions in the original equation for extraneous roots.
Example
$\sqrt{2x - 1} = x - 2$. Square both sides: $2x - 1 = (x-2)^2 = x^2 - 4x + 4$. Rearrange: $x^2 - 6x + 5 = 0 = (x-1)(x-5)$. Solutions: $x = 1$ and $x = 5$. Check $x = 1$: $\sqrt{1} = 1$ and $1 - 2 = -1$. No match. Extraneous. Check $x = 5$: $\sqrt{9} = 3$ and $5 - 2 = 3$. Correct. Only $x = 5$.
Key Insight
Squaring both sides of an equation is not an equivalence transformation - it can introduce false solutions. Extraneous solutions arise because squaring removes sign information.
Definition
Solving a radical equation involves isolating a radical of the form $f(x)^{1/n} = g(x)$ and raising both sides to the n-th power, yielding $f(x) = g(x)^n$. This is a surjection (not injection) when $n$ is even, potentially introducing extraneous roots from the negative branch of $g(x)$. The complete solution set is the intersection of the solution set of $f(x) = g(x)^n$ with the domain constraints ($g(x) \ge 0$ for even $n$, to match the principal root convention).
Example
$\sqrt{x} = -3$ has no solution over $\mathbb{R}$ (even though squaring gives $x = 9$, since $\sqrt{9} = 3$, not $-3$). The principal square root is always non-negative.
Key Insight
The issue with radical equations is that raising to an even power is a many-to-one operation. In complex analysis, the multi-valued nature of n-th roots (there are $n$ distinct n-th roots of any non-zero complex number) is handled rigorously via Riemann surfaces.