Question

Find the largest real number $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$.

Answer 1

We need x with x^2+3x+c = -2, i.e. x^2+3x+(c+2)=0. This has a real solution iff the discriminant is ≥0:
9 - 4(c+2) ≥ 0 → 1 - 4c ≥ 0 → c ≤ 1/4.

Thus the largest such c is 1/4. (For c = 1/4 the unique solution is x = -3/2, giving f(-3/2) = -2.)