2009 AMC 12A Problem 23

Functions f and g are quadratic, g(x) = - f(100 - x), and the graph of g contains the vertex of the graph of f. The four x-intercepts on the two graphs have x-coordinates x_1, x_2, x_3, and x_4, in increasing order, and x_3 - x_2 = 150. Then x_4 - x_1 = m + n\sqrt p, where m, n, and p are positive integers, and p is not divisible by the square of any prime. What is m + n + p?

\textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \textbf{(D)}\ 752 \qquad \textbf{(E)}\ 802

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