We describe how late Babylonian pupils (circa $500$ BCE) were shown how to solve the system of equations $a+b=S$, $ab=P$. We use the computer screen instead of incisions on a clay tablet, so our solution will be much shorter-lived than theirs. To make the derivation more familiar, we use algebraic notation that came a couple of thousand years later. But we give a correct description of the algorithm that was taught.
We have $(a+b)^2=S^2$. Subtract $4ab$. We get that $(a-b)^2=S^2-4P$. Take the square root(s). We get $a-b=\pm \sqrt{S^2-4P}$ (but there were no negative numbers back then either, those were the good old days). Now we know $a+b$ and $a-b$. Add and subtract, divide by $2$, to find $a$ and $b$.
So we have solved a "quadratic equation" without using the quadratic formula, indeed without writing down the equation. You will recognize the procedure as a slightly hidden completing the square. Which reminds me, for al-Khwarizmi and many years after that, completing the square meant completing a geometric figure made up of a square with a square bite taken out of a corner to a real geometric square.