Quadratic equation with parameter (what method to use to obtain correct answer)

Question

For which values of the parameter $k$ does the equation $(2k-5)x^2-2(k-1)x+3=0$ have exactly one real answer?

From my work I determined $k = 4$, but the answer is $[k=4 \lor k=\frac{5}{2}]$. The symbol $\lor$ I believe is supposed to indicate (OR). Obviously inputting $\frac{5}{2}$ makes the $a$ coefficient ($ax^2$) equal to $0$, therefore it's not a quadratic equation anymore, but I am not sure how I would have gotten to that answer (is this a trick question, or is there a procedure I need to follow when solving these to actually get the answer right every time?)

Answer 1

The question is poorly worded. It stipulates real equal answers, plural, and D = 0, which would seem to exclude their answer of $\frac{5}{2}$, because if you have only one answer, you might argue that it is "trivially" equal to itself, but the discriminant D is a feature of a quadratic equation, not a linear one.

And yes that $\lor$ is the logical OR operator.

Edit: The reworded question asks for "exactly one real answer" and that can be achieved by either making the equation linear or by making the two roots of the quadratic match, so their answer makes sense now. In general it is good to look for edge cases and loopholes: "what if this is zero (negative, less than 1, etc?" In particular for polynomials a common edge case is if the leading coefficient becomes zero; you have to watch out for that in many proofs involving polynomials.