<s>[INST] How to solve for X in Cubic poynomial I've been given a Polynomial (Cubic)
$$k=\frac16x\cdot(x+1)\cdot(2x+1)$$
If $k$ is given, is there any way to solve for $x$?
 [/INST] We solve the problem in a very special case only, where $x$ happens to be a positive integer. 
Since $\frac{x(x+1)(2x+1)}{6}\gt \frac{x^3}{3}$, we have $x^3\gt 3k$, and therefore $x\gt (3k)^{1/3}$. 
Note that $2(x+1)^3=2x^3+6x^2+6x+2$ while $x(x+1)(2x+1)=2x^3+3x^2+x$. It follows that if $x$ is a positive integer then $\frac{x(x+1)(2x+1)}{6}\lt \frac{(x+1)^3}{3}$, and therefore $x+1\gt (3k)^{1/3}$. 
It follows that $x=\lfloor (3k)^{1/3}\rfloor$. 
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