What is the solution for an inverse third order equation when solving for X?

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To solve the inverse third order equation for X, start by multiplying the equation y = y_{0} + (a/x) + (b/x^2) + (c/x^3) by x^3 to eliminate the denominators. This transforms the equation into a cubic polynomial of degree three. The roots of this polynomial can be found using Cardano's method, which is a standard approach for solving cubic equations. The discussion highlights the importance of maintaining the integrity of the data when swapping axes, as it can significantly affect the R² value. Understanding these steps is crucial for independently calculating both Y and X values.
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I have a question about solving a formula.

I have a set of data which I have fit to an inverse third order equation, if I swap the axes (use X as Y and vise versa), R2 suffers dramatically but I would like to calculate both Y and X values independantly. I am wondering what the solution for the following equation is when cast to calculate X. My algebra is a little rusty and I am not sure where to start.

y = y_{0}+\frac{a}{x}+\frac{b}{x^{2}}+\frac{c}{x^{3}}

Assuming I have values for a,b,c, and yo, how do I solve this for x?
 
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