We want to find $a_1$ and $a_2$ using only $p$ and $q$
$$
x^2 - \underbrace{(a_1+a_2)}_{p}x + \underbrace{a_1a_2}_{q} = (x-a_1)(x-a_2).
$$
With
$$
q= a_1a_2= (m-d)(m+d) = m^2 - d^2,
$$
we immediately get the p-q formula
$$
\begin{aligned}
a_{1/2}
&= m \pm d \\
&= m \pm \sqrt{m^2 - q}\\
&= \frac{p}{2} \pm \sqrt{\left(\frac{p}{2}\right)^2 - q}
\end{aligned}
$$
Source: 3blue1brown