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 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

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

 Polynomial roots of the fifth degree 0.74954440929537-0.92660132700825i -0.42604117629241-0.69141199730656i -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 0.80517978551219-0.90690579788299i -0.42780028378999-0.63253712529931i -1.0695749012912+0.51597635530179i -0.23323335872174+0.95142805026712i 0.92542875829085+0.072038517613355i
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