A random quadratic

Idea. The condition on the coefficients is a region in the unit square, and its area is the probability.

Step 1: the region. The roots are real when $B^2\ge 4C$, that is $C\le \frac{B^2}{4}$. Since $B\le 1$, the bound $\frac{B^2}{4}$ is at most $\frac{1}{4}$ and always inside $[0,1]$.

Step 2: the area. $$\mathbb{P}\left(C\le \frac{B^2}{4}\right)={\int }_0^1\frac{b^2}{4}db=\frac{1}{12}.$$

Variant. For $x^2+2Bx+C=0$ the condition becomes $C\le B^2$ and the answer is $\frac{1}{3}$. A change of convention in the statement changes the answer by a factor of $4$, so read the statement carefully in an interview.