quadratics - Intuitive meaning behind the Discriminant

While I was applying the quadratic formula, it just suddenly dawned to me that the quadratic formula was basically finding the vertex and adding or subtracting (the plusminus) half the distance between the roots. $\frac{-b}{2a}$ represents the vertex and the $\pm \frac{\sqrt{\triangle}}{2a}$ represents half the distance between the roots. I understand why $-\frac{b}{2a}$ is the xcoor of the vertex, it intuitively makes sense. But why does the $\frac{\sqrt{\triangle}}{2a}$ equal half the distance between the roots? i know how to prove it but I dont why it intuitively works.
Edit: I understand why the $\frac{\sqrt{\triangle}}{2a}$ is half the distance from the vertex but why is the discriminant B^2 - 4ac. I know thats what happens when you complete the square but when you use algebra you often lose the intuitive sense of what's happening.