I'm not sure if this SE site is the best one for this question, so let me know where it should be moved to if you think it doesn't belong here.
After learning about the quadratic formula, I'm interested in visualizing how the nature of the roots of a quadratic equation, $ax^2+bx+c$ change with respect its discriminant. I'd like to visualize the discrete nature of the roots of a quadratic equation over all combinations of $a, b, c$ in $b^2-4ac$, i.e. whether the roots are complex, equal, unequal, rational, or irrational, etc. I'm not sure I fully grasp the nature of my question itself and how to get a mathematically sound answer to it.
This is interesting to me because it deals with discrete properties emerging from continuous ones. I'm wondering what is the best way of visualizing/answering this – should I do random sampling? where can I read more about problems of this nature (i.e. discrete properties on continuous variables)?