This question is about the methods to solve polynomials of the fourth degree with the form $ax^4 + bx + c = 0$.
I am currently creating an algorithm that needs to calculate the solution of the equation $a T^4 + bT + c = 0$, with $T$ being the temperature. Obviously, temperatures are positive reals, so I need the only positive real root.
In the algorithm of MATLAB, I use a function to solve any polynomial; but I would like to compare it to a direct formula. (Empirically, there is always one and only one positive real root, but I haven't proved it) So I went to look up for the equation of the solutions on Wolfram Alpha to get the general root expressions and try to compute it directly.
However, I always ended up with complex results, I don't know exactly why. Also the solution provided by Wolfram Alpha doesn't show where the square root sign ends, so I am pretty sure I messed that up. So what is the general expression for the equation $ax^4 + bx + c = 0$ (an expression in which it is clear where the operators start and end; or using temporary variables to calculate it)?
Finally, I tried to check some methods to solve fourth-degree polynomials and tried to use the Ferrari method, but I didn't end up getting a real root.
Are there methods that don't required "advanced" calculating methods (MATLAB root function), but ones based on elementary operations?