Equation of the fifth degree online: a special case

In this material, one of the solutions of the fifth-degree equation of a particular form is considered. Today, the Abel-Ruffinni theorem states that there is no general solution to a fifth-degree equation that can be expressed in a finite number of calculations using arithmetic operations, exponentiation and root extraction. 

At least two assumptions can be made from this theorem:

- there are particular types of the equation of the 5th degree, the roots of which can be found by substitution into a certain (inherently finite) formula

- if it is impossible to find a general solution, where "a finite number of operations", then theoretically it is possible to search for a general solution using functions where "an infinite number of operations

I will offer you an online solution to an equation of the fifth degree of this kind

\(x^5+ax^3+a^2/5x+b\)

To do this, we need to calculate two auxiliary parameters  \(F\) and \(T\)

\(\cfrac{-i*c}{2b}\sqrt{\cfrac{125}{a}}=F\)

\(\sqrt{\cfrac{4a}{5}}=T\)

After that we can find all the roots of such an equation.

Works in the case when the original data are complex numbers.

Recall that the discriminant of such a polynomial is

\(\cfrac{16 a^{10} + 25000 a^5 b^2 + 9765625 b^4}{3125}\)

Examples

\(x^5+ix^3-\cfrac{x}{5}+i=0\)

But not without a fly in the ointment. There are errors in calculations, or rather in the choice of the root sign. Here's one example. 

\(x^5+(tan(2+i))x^3+\cfrac{(tan(2+i))^2}{5}x​+(i)=0\)

On this equation, despite the fact that all values ​​are the same, the sign must be changed to the opposite. Why and what criterion, I have not yet understood.