Solve Algebraic Equation: ax^(n-1) - x + 1 - a = 0

[Wuberdall]

Hi,

I'm currently studying percolation theory and here I'm stuck on the this "simple" algebraic equation
$$ax^{n-1} - x +1 - a = 0$$
clearly, the trivial solution x = 1 solves it. But I'm told that
$$x = 1 - \frac{2a(n-1)-2}{a(n-2)(n-1)}$$
is another solution. This makes we wonder, if the equation can be put into a simpler form (e.g. quadratic or cubic) — even though I haven't succeed to.

Can any of you help ??

Thaks in advance.

 

[FactChecker]

I don't think that the statement is true. With a=4, n=4, I get x=1/12 and f(1/12) = -1331/432

 

[Stephen Tashi]

clearly, the trivial solution x = 1 solves it.

Then you can obtain an equation in a lower power of x by dividing the original equation by (x-1). Try that for a particular case like n = 3.