Squaring both sides invents answers
Graph both sides and there is nothing to check — a graph cannot invent a crossing.
1:50 · Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables
Worked in this video
√(x + 7) = x − 5
What is the solution set?
Squaring both sides gives x² − 11x + 18 = 0, so x = 2 or x = 9 — but x = 2 fails the original equation, because √9 is 3 while 2 − 5 is −3. The radical sign means the principal root, so it is never negative, and squaring invented that second answer. In Desmos, type y = √(x + 7) and y = x − 5 instead. The curves cross exactly once, at x = 9, and there is nothing to check afterwards: a graph cannot invent a crossing that is not there.
Answer: {9}