Solve simple rational and radical equations and identify extraneous solutions.
Solve a radical equation with radicals on both sides and check the domain
Problem
Solve \(\sqrt{x~+~5}~=~\sqrt{2x~-~1}\) and verify the common domain.
Big Picture
What this problem is really about
When square roots appear on both sides, first find the inputs allowed by both radicands; the common domain is the intersection of those conditions. On that domain, squaring compares the radicands and produces a simpler equation. Treat its result as a candidate until substitution confirms that both original principal square roots are defined and equal.
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