1963 AHSME Problem 24

Consider equations of the form x^2 + bx + c = 0. How many such equations have real roots and have coefficients b and c selected from the set of integers \{1,2,3, 4, 5,6\}?

\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 16

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