jmc

Seeing similar expressions in multiple places, we make a substitution: let $y=x^2-10x-45$. Then we have the equation $$\frac{1}{y+16}+\frac{1}{y}-\frac{2}{y-24}=0.$$Multiplying by $(y+16)(y)(y-24)$ to clear denominators, we have $$y(y-24)+(y+16)(y-24)-2y(y+16)=0$$or $$-64(y+6)=0.$$Thus $y=-6$, so $-6=x^2-10x-45$ or $$x^2-10x-39=0.$$This factors as $(x-13)(x+3)=0$, so the positive solution is $x=\boxed{13}$.