The discriminant of a quadratic

Relation to roots. By Vieta, $x_1+x_2=-b/a$, $x_1x_2=c/a$. Then
$$(x_1-x_2)^2=(x_1+x_2)^2-4x_1x_2=\frac{b^2}{a^2}-\frac{4c}{a}=\frac{b^2-4ac}{a^2}=\frac{\Delta}{a^2}.$$
So $\Delta=a^2(x_1-x_2)^2$.

Sign of $\Delta$.
- If $\Delta>0$: two distinct real roots.
- If $\Delta=0$: one double real root.
- If $\Delta<0$: two complex conjugate roots.

Conclusion. The sign of $\Delta$ determines the nature of the roots, and $|\Delta|^{1/2}=|a||x_1-x_2|$ is the distance between the roots (in $\mathbb{R}$).