Question: I am wanting to reverse-engineer an equation due to a previously unnoticed rounding error and I am trying to determine if it's possible to solve for x. In theory, the values of z and y would always be known since they have been recorded...x was rounded after the initial calculations were done. Here is the equation.

$\ z = 6.5yx^2 - 3.25x^3 $

As it has been 13 years since my last math class, I cannot recall if anything can be done when an additional variable, in this case "y", is present. Is it possible to solve this equation for x?

Answer: 
Using whole numbers and simplifying, the equation becomes
$$x^3-2 x^2 y+\frac{4 }{13}z=0$$ To make it more workable, let $a=-2y$ and $b=\frac{4 }{13}z$ and solve $$x^3+ax^2+b=0$$ Make it a depressed cubic equation using $x+\frac a3=t$ to get
$$t^3-\frac{a^2 }{3}t+\frac{2 a^3}{27}+b=0$$ Let $p=-\frac{a^2 }{3}$ and $q=\frac{2 a^3}{27}+b$ to end with
$$t^3+p t+q=0$$ and now, use the formulae given [here](https://en.wikipedia.org/wiki/Cubic_function) starting at equation $(2)$.

If you know that for any $(y,z)$ there is only one real root (if I am not mistaken, this would imply $z \left(27 z-104 y^3\right) >0$), then the problem would be simpler using the hyperbolic solution.
This is a cubic with solutions of the form:

$$x=-(0.00190095\, +0.00329253 i) \sqrt[3]{1678.36 \sqrt{2.81688\times 10^6 z^2-1.0985\times 10^7 y^3 z}+5.4925\times 10^6
 y^3-2.81688\times 10^6 z}-\frac{(59.1765\, -102.497 i) y^2}{\sqrt[3]{1678.36 \sqrt{2.81688\times 10^6
 z^2-1.0985\times 10^7 y^3 z}+5.4925\times 10^6 y^3-2.81688\times 10^6 z}}+0.670795 y$$