You can interrogate a quadratic's roots without ever finding them. The two roots of $ax^2 + bx + c = 0$ always add to $S = -\dfrac{b}{a}$ — S equals minus b over a — and multiply to $P = \dfrac{c}{a}$ — P equals c over a. The minus rides the SUM; the product travels unsigned. This is Vieta's pair, and it is the exam hall's best five seconds: solved roots that fail either check are wrong, full stop.

And here is the stern part, in your favour: two numbers with the right sum AND the right product ARE the roots — pass both checks and the audit is conclusive. One nugget to carry back: the vertex sits at $-\tfrac{b}{2a}$, which is exactly HALF the sum — the turning point is the average of the landings, which is what mirror symmetry has been saying all along.