= Roots from a completed square
Setting $a(x-h)^2+k=0$ gives $(x-h)^2=-k/a$. A positive $-k/a$ gives two distinct <roots of a polynomial> in the <real numbers>; zero gives a repeated <root of a polynomial> at $h$; a negative value gives a pair of <complex conjugate> <roots of a polynomial>. For the downward-opening example, the two <roots of a polynomial> are $2\pm\sqrt{3/2}$, approximately $0.775255$ and $3.224745$.
The <discriminant> $\Delta=b^2-4ac$ packages the same calculation, because
$$
k=-\frac{\Delta}{4a},\qquad (x-h)^2=\frac{\Delta}{4a^2}.
$$
Since $4a^2>0$, the sign of $\Delta$ determines which of the three cases occurs. This derives the <quadratic formula> by a <polynomial identity>, and explains its structure: first shift to $h$, then take a <square root>. For the upward-opening example, $(x+3)^2=-4$ gives the <complex conjugate> <roots of a polynomial> $-3\pm2i$.
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