2003 AMC 12A Problem 25

Let f(x)= \sqrt{ax^2+bx}. For how many real values of a is there at least one positive value of b for which the domain of f and the range of f are the same set?

\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ \mathrm{infinitely \ many} }

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.