polynomials - Why is it that the roots of a quadratic is closest when the discriminant is smallest (but non-zero)?

My teacher told me earlier today that the roots of a quadratic are the closest when the discriminant is smallest, but non-zero. I wasn't able to understand why that is the case, however.

Is it correct to say that the discriminant is the distance between the roots? And since you can't have negative distance, that's why the roots are imaginary when the discriminant is negative?

Answer

If the roots coincide the discriminant is $0$ ($(x-r)^2=x^2-2rx+r^2$ has discriminant $0$) so it is natural to imagine that a very small discriminant means that the roots are close. However, the relation between the difference and the discriminant is not as simple as you might like.

If the quadratic is $ax^2+bx+c$ with discriminant $\Delta =b^2-4ac$ then the roots are $r_{\pm}=\frac {-b\pm \sqrt {\Delta}}{2a}$ so the difference is

$$r_+-r_-=\frac {\sqrt {\Delta}}a$$

Thus

$$\boxed {\Delta = a^2(r_+-r_-)^2}$$