Question:

Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Multiply both sides by $(x-1)$ (noting $x \neq 1$): $x(x-1) - 3 - (x-1) = x$ $x^2 - x - 3 - x + 1 = x$ $x^2 - 3x - 2 = 0$ Discriminant $= 9 + 8 = 17 > 0$, giving two roots. But check if either equals 1: substituting shows one root is extraneous (or the original simplifies differently). Per MFA033 the answer is one valid root. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.