$2 \cdot 121$ In the equation $\frac{x(x-1)-(m+1)}{(x-1)(m-1)}=\frac{x}{m}$, if the roots are equal, then:
(A) $m=1$.
(B) $m=\frac{1}{2}$.
(C) $m=0$.
(D) $m=-1$.
(E) $m=-\frac{1}{2}$.
(2nd American High School Mathematics Examination, 1951)
［Solution］From the given equation, we transform it to get
$$
\frac{x(x-1)-(m+1)}{m-1}=\frac{x(x-1)}{m},
$$

By the property of proportions, we get $\frac{m+1}{1}=\frac{x(x-1)}{m}$,
which simplifies to $(m+1) m=x(x-1)$, yielding $x=m+1$ or $-m$.
If $m+1=-m$, then $m=-\frac{1}{2}$, which is the condition for the equation to have equal roots. Therefore, the answer is $(E)$.