Since I don't have enough reputation to comment, I am asking this question again.
I cannot understand why we can assume that the quadratic has real roots and then say $D\ge0$. The answer states that $b^2 - 4ac = (b + 2ax)^2$ and since this is a squared term it satisfies for all $x$ belonging to $\mathbb R$. But that is the question we began with, how do we know that $x$ is real. Plugging a non-real value of $x$ does not satisfy the relation of $D$.