I have been thinking about this:

When we graph $\displaystyle y = ax^2 + bx + c$, we get a parabola opening upwards or downwards. It will intersect/touch the x-axis if it has real roots and we know the value of the roots from the points of intersection. When the equation has complex roots, the parabola naturally doesn't intersect with the x-axis. Is it possible for us to locate those complex roots? For eg., if the roots are $\displaystyle 1 \pm \sqrt{5}i$, is it possible to discern from the graph the location of those roots? My professor says Yes. And I fail to see how. (And no, this isn't about the Argand plane. We are talking about the real x-y axes)