Values of quadratic function and its discriminant

The question might be trivial, but I confused in one step which I tried to mention.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be $f(x)=ax^2+bx+c$ where $a\neq 0$.

Suppose that $f(x)\ge 0$ for all $x\in\mathbb{R}$.

I do not understand, why discriminant of $f$ (i.e. $b^2-4ac$) should be negative?

It might be trivial, but I got confused with values of function and discriminant; discriminant tells only possible roots of $f$; how the values of $f$ pose condition on discriminant of it?