[math-fun] Geometric significance of algebraic discriminant ?

I recall learning a bunch of algebra about the discriminant, which becomes zero when there are coincident roots. https://en.wikipedia.org/wiki/Discriminant Has someone come up with geometric insights about this particular formula ? In the case of a quadratic, the formula is (x1-x2)^2, but this isn't the real number |x1-x2|^2. Perhaps the norm of the discriminant (DD*) is more important? What about the discriminant of the cubic ? Shouldn't this say something interesting about the triangle in the complex plane formed by the roots?