MathNet 2016-part-04 — 184 problems (EN–VI) Question 131

A quadratic polynomial $f(x) = ax^2 + bx + c$ has no real roots. It is given that $b$ is a rational number, and exactly one of $c$ and $f(c)$ is a rational number. Is it possible for the discriminant of $f(x)$ to be a rational number?