Quartic equation roots

Returns all the roots of a quartic equation with a real or complex number coefficient.

$a\ast x^4+b\ast x^3+c\ast x^2+d\ast x+e=0$

Note 1: It should be $a\ne 0$

Note 2: If the coefficient is a real number, the real number will be written in the first box, the second box will be zero, if the coefficient is a complex number, the real part of the number will be written in the first box, and the virtual or image part will be written in the second box.

Equation coefficients :

$a=$+ $i$

$b=$+ $i$

$c=$+ $i$

$d=$+ $i$

$e=$+ $i$

All the roots of the fourth degree equation are found by doing the following operations sequentially.
Equation coefficients divided by $a$. $B={\displaystyle \frac{b}{a}}$, $C={\displaystyle \frac{c}{a}}$, $D={\displaystyle \frac{d}{a}}$, $E={\displaystyle \frac{e}{a}}$ values ​​are found. The following $\alpha$ and $\beta$ values ​​are found.
$\alpha =27E{B}^2-9BCD+2C^3-72EC+27D^2$
$\beta =-3BD+C^2+12E$

The following $\delta$, ${\xi }_1$, ${\xi }_2$, ${\epsilon }_1$, ${\epsilon }_2$, $\mathrm{\Delta }$, ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ values ​​are calculated, respectively.

$\delta ={\displaystyle \sqrt[{\displaystyle 3}]{\sqrt{{\alpha }^2-4{\displaystyle {\beta }^3}}+{\displaystyle \alpha }}}$

${\xi }_1={\displaystyle \frac{{\displaystyle \delta }}{{\displaystyle 3{\displaystyle \sqrt[3]{{\displaystyle 2}}}}}+{\displaystyle \frac{{\displaystyle \sqrt[3]{{\displaystyle 2}}{\displaystyle \beta }}}{3{\displaystyle \delta }}}}$

${\xi }_2={\displaystyle \frac{B^2}{4}-{\displaystyle \frac{2C}{3}}}$

${\displaystyle {\epsilon }_1={\displaystyle \frac{-B^3+4BC-8D}{4{\displaystyle \sqrt{{\displaystyle {\xi }_1+{\displaystyle {\xi }_2}}}}}}}$

${\displaystyle {\epsilon }_2={\displaystyle \frac{-{\displaystyle \delta }}{3{\displaystyle \sqrt[3]{2}}}-{\displaystyle \frac{\sqrt[3]{2}\beta }{3\delta }+{\displaystyle \frac{B^2}{2}}}}}$

${\displaystyle \mathrm{\Delta }={\displaystyle \frac{1}{2}\sqrt{{\xi }_1+{\xi }_2}}}$
${\displaystyle {\mathrm{\Delta }}_1={\displaystyle \frac{1}{2}\sqrt{{\displaystyle {\epsilon }_2-{\displaystyle {\epsilon }_1-{\displaystyle \frac{4C}{3}}}}}}}$
${\displaystyle {\mathrm{\Delta }}_2={\displaystyle \frac{1}{2}\sqrt{{\displaystyle {\epsilon }_2+{\displaystyle {\epsilon }_1-{\displaystyle \frac{4C}{3}}}}}}}$

Roots of the equation
Root 1 :       ${\displaystyle {\varkappa }_1=-{\displaystyle {\displaystyle \mathrm{\Delta }-{\displaystyle {\mathrm{\Delta }}_1-{\displaystyle \frac{B}{4}}}}}}$

Root 2 :       ${\displaystyle {\varkappa }_2=-{\displaystyle \mathrm{\Delta }+{\displaystyle {\mathrm{\Delta }}_1-{\displaystyle \frac{B}{4}}}}}$

Root 3 :       ${\displaystyle {\varkappa }_3={\displaystyle \mathrm{\Delta }-{\displaystyle {\mathrm{\Delta }}_2-{\displaystyle \frac{B}{4}}}}}$

Root 4 :       ${\displaystyle {\varkappa }_4={\displaystyle \mathrm{\Delta }+{\displaystyle {\mathrm{\Delta }}_2-{\displaystyle \frac{B}{4}}}}}$