You can interrogate a quadratic's roots without ever finding them. Whatever the two roots of $ax^2 + bx + c = 0$ turn out to be, they 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. $S$ is the sum of the two roots and $P$ their product; the minus rides the sum, and the product travels unsigned. This is Vieta's pair, and it is the best five seconds you will spend in an exam hall: roots that fail either check are wrong, full stop. The stern half is in your favour — two numbers with the right sum AND the right product ARE the roots. Pass both and the audit is conclusive.

Now the line that makes this an optimization lesson. The vertex sits at $-\dfrac{b}{2a}$, which is exactly HALF of $S$. So the optimum is the AVERAGE of the two break-even points — hand someone the two outputs at which a business washes its face, and they can name the most profitable output without seeing a single coefficient. Write the roots as $r_1$ and $r_2$, with $r_1$ the smaller and $r_2$ the larger, and $h = \dfrac{r_1 + r_2}{2}$. Mirror symmetry has been saying this all along.

The third tool here builds the vertex from scratch rather than reading it off. $x^2 + bx$ is an unfinished square; add $k = \left(\tfrac{b}{2}\right)^2$ — k equals b over two, squared, where $b$ is the coefficient of the first-power term and $k$ the constant that completes it — and it snaps shut into $\left(x + \tfrac{b}{2}\right)^2$. Half FIRST, square second; reversing the order misses by a factor of four. Completing the square is $-\dfrac{b}{2a}$ in slow motion, and doing it twice will make you believe the vertex formula forever.