§ Galois theory perspective of the quadratic equation

I found this quite delightful the first time I saw it, so I wanted to record it ever since. Let $x^2 + bx + c$ be a quadratic. Now to apply galois theory, we first equate it to the roots:
\begin{aligned} &x^2 + bx + c = (x - p)(x-q) &x^2 + bx + c = x^2 - x(p + q) + pq &-(p + q) = b; pq = c \end{aligned}
We want to extract the values of $b$ and $c$ from this. To do so, consider the symmetric functions:
$(p + q)^2 = b^2 (p - q)^2 = (p + q)^2 - 4pq = b^2 - 4c$
Hence we get that
$p - q = \pm\sqrt{b^2 - 4c}$
From this, we can solve for $p, q$, giving us:
$p = ((p + q) + (p - q))/2 = (-b \pm \sqrt{b^2 - 4c})/2$