$1^2 + 2^2 + 3^2 + 4^2 + \cdots + N^2 = S$
Given $S$
How to find $N$.
The Formula to Find $S$ from $N$ is:
$S = \frac{N(N+1)(2N+1)}6$
so this gives me a cubic equation:
$2N^3 + 3N^2 + N = 6S$
So, how to solve this, because in synthetic division we have to know one of the roots but many times its not possible.
ps: I know I could have asked the last part directly but I gave the full scenario so that if there is some other method then that is also answered.
$N^3<3S<(N+1)^3$, and $N$ is an integer.