Two values of the private equation of the 5th degree 

The given equation 
Polynomial roots of the fifth degree 
In this material, one of the solutions of the fifthdegree equation of a particular form is considered. Today, the AbelRuffinni theorem states that there is no general solution to a fifthdegree 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
\(x^5+ix^3\cfrac{x}{5}+i=0\)
Polynomial roots of the fifth degree 
0.749544409295370.92660132700825i

0.426041176292410.69141199730656i

1.012852336851+0.49928521244333i

0.19993599346623+0.99998722867676i

0.88928509731439+0.11874088319475i

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.
Polynomial roots of the fifth degree 
0.805179785512190.90690579788299i

0.427800283789990.63253712529931i

1.0695749012912+0.51597635530179i

0.23323335872174+0.95142805026712i

0.92542875829085+0.072038517613355i
