if $x_{1},x_{2} $ is $nx-x^n=a$ two roots, show that $|x_{1}-x_{2}|<\dfrac{a}{1-n}+2$ 
Assmue that $n$ be postive integers,and $a$is real number,and the equation $$nx-x^n=a$$ has postive real roots $x_{1},x_{2}$,show that
  $$|x_{1}-x_{2}|<\dfrac{a}{1-n}+2$$

By condition, I showed that
$$(x_{1}-x_{2})[n-(x^{n-1}_{1}+x^{n-2}_{1}x_{2}+\cdots+x^{n-1}_{2})]=0$$
But I don't know what to do then. Please help, Thanks.

Let $f(x)=nx-x^n-a$. Then by Rolle's theorem we get, $\exists c\in [x_1,x_2]$ such that $f'(c)=0\implies c=1$ assuming $n>1$. Then, $\exists \delta>0$ such that $x_1=1-\delta, x_2=1+\delta$. Now, $$nx_2-x_2^n=a\implies n(1+\delta)-(1+\delta)^n=a\\\implies a=(1+\delta)(n-(1+\delta)^{n-1})\\\implies a<(1+\delta)(n-(1+(n-1)\delta))\\\implies a< (n-1)(1-\delta^2)\\\implies 2(\delta-1)<(\delta^2-1)<\frac{a}{1-n}\\\implies |x_1-x_2|=2\delta<\frac{a}{1-n}+2 $$
