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?
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 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.