$\frac{1}{m}+\frac{1}{n}-\frac{1}{x}=\frac{1}{m+n-x} \\ \frac{nx+mx-mn}{mnx}=\frac{1}{m+n-x} \\ mnx+n^2x-nx^2+m^2x+mnx-mx^2-m^2n-mn^2+mnx=mnx \\ -x^2(m+n)+x(m+n)^2-mn(m+n)=0 \\ -(m+n)(x^2-x(m+n)+mn)=0 \\ m+n \neq 0 \\ \frac{x_1+x_2}{x_1 x_2}=\frac{-\frac{b}{a}}{\frac{c}{a}}=-\frac{b}{c}=\frac{m+n}{mn}$