2015 AIME II Problem 7

Triangle A B C has side lengths A B=12, B C=25, and C A=17. Rectangle P Q R S has vertex P on \overline{A B}, vertex Q on \overline{A C}, and vertices R and S on \overline{B C}. In terms of the side length P Q=w, the area of P Q R S can be expressed as the quadratic polynomial

\operatorname{Area}(P Q R S)=\alpha w-\beta \cdot w^{2}

Then the coefficient \beta=\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.