Smallest nonnegative root of this polynomial?

Question

Apparently this should be easy to compute. Suppose that $0 < c < 1/2$ and $d > 0$ such that $d + c = 1$. Then what is the smallest nonnegative root of $p(t) = 2(c + dt^2)^n - t$, where $n$ is a positive integer?

Answer 1

With the change of variables $x = c + d t$, the equation becomes $$ x - 2 d x^n = c$$ Although for $n \ge 5$ there is no solution in radicals, there is a solution as a generalized hypergeometric series in $d$, converging for sufficiently small $d$.