# Solving $a x^4 + b x + c = 0$

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?

• this link may be helpful: math.stackexchange.com/questions/785/… – Sujit Bhattacharyya Nov 15 '18 at 11:52
• Computing the discriminant and using Descartes' Rule of Signs shows that unless $c = 0$, this equation either has $0$ or $2$ real roots, according to whether $256 a c^3 - 27 b^4$ is positive or negative, respectively. (If the discriminant is zero, either is possible.) – Travis Nov 15 '18 at 12:28
• At any rate, the general formula is somewhat unpleasant, and probably for your (empirical) setting a numerical solution is easier and more reasonable besides. – Travis Nov 15 '18 at 12:29