Given variable $m$, how do I find zeros of a polynomial in terms of $m$?

Question

This is a summation question about a finite series with sum $m$. I'm trying to write a computer program that takes in a given integer $m$ (which represents the sum of a series) and outputs the number of terms — or in this case, cubes — that are in the series.

Here is the notation:

$\sum_{k=0}^{n} k^3 = m$

I know that

$\sum_{k=0}^{n} k^3 = \frac{n^2(n+1)^2}{4}$

so,

$m = \frac{n^2(n+1)^2}{4}$

How do I find the zeros in terms of $m$? or the formula for $n$ in terms of $m$? (i.e. $n =$ ???)

Answer 1

from $$m=\frac{n^2(n+1)^2}{4}$$ from here we get $$4m=n^2(n+1)^2$$ or $$\sqrt{4m}=\pm n(n+1)$$ for $+$ we get $$n^2+n-\sqrt{4m}$$ $$n_{1,2}=-\frac{1}{2}\pm\sqrt{\frac{1}{4}+\sqrt{4m}}$$ can you finish?