If the equation $\displaystyle \frac{x^{2}-bx}{ax-c}=\frac{m-1}{m+1}$ has roots equal in magnitude but opposite in sign, then $m=$

For the equation to have roots equal in magnitude but opposite in sign, the sum of the roots must be zero. Given the equation (x^2 - bx) / (ax - c) = (m-1) / (m+1), cross-multiply to get (m+1)(x^2 - bx) = (m-1)(ax - c). Expanding and rearranging into a quadratic form ax^2 + Bx + C = 0, the coefficient of x must be zero for the sum of roots to be zero.