solution of an algebraic equation?

Question

There is an algebraic equation like $ax^{2n-2}-bx^{2n-4}+c=0$, where $a,b,c>0$ and $n$ is an integer with $n\geq3$. What are the solutions of this equation or the properties of its solutions?

Answer 1

$p(x) = a x^{2n-2} - b x^{2n-4} + c$ is a polynomial of degree $2n-2$. Since it's even, its roots are symmetric under $x \to -x$. Its derivative is $0$ at $x=0$ and $x = \pm \sqrt{\frac{b(n-2)}{a(n-1)}}$. The latter are also roots of $p(x)$ (and thus double roots) if $$c = (n-1)^{n+1} (n-2)^{n-2} a^{-n+2} b^{n-1}$$ Otherwise, all roots are simple.

While in general there is no solution in radicals, there are e.g. power series solutions, which may lead to hypergeometric functions.

Answer 2

We use a change of variables $y = x^2$ and $k = n-1$. Then the equation becomes $ay^k - by^(k-1) + c = 0$. Then $y = x^2$, so $x = \pm \sqrt{y}$. There are no general solutions to this in terms of $a,b,c,(k+1)$, I believe.

What sort of properties are you thinking about? Of course, we could say the sum of the roots of $f(y)$ are $\frac{b}{a}$ by Vieta's, etc. but it is hard to give a complete answer without any context or direction.