Squaring both sides invents answers
Graph both sides: the graph adds no extra answers to throw away. Then check the one it shows.
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 once, at x = 9 — zoom out to make sure there is no second crossing off screen. The graph adds no extra crossing for you to throw away, but still check the one it shows in the original: √16 = 4 and 9 − 5 = 4.
Answer: {9}