2013 AMC 12A Problem 12

The angles in a particular triangle are in arithmetic progression, and the side lengths are 4,5,x. The sum of the possible values of x equals a+\sqrt{b}+\sqrt{c} where a, b, and c are positive integers. What is a+b+c?

\textbf{(A)}\ 36\qquad\textbf{(B)}\ 38\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 44

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