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Quartic equation roots

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


Note 1: It should be \(a\ne0\)

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.
\(\alpha= 27 E B^2 - 9 B C D + 2 C^3 - 72 E C + 27 D^2 \)
\(\beta=-3 B D + C^2 + 12 E\)

The following \(\delta\), \(\xi_1\), \(\xi_2\), \(\varepsilon_1\), \(\varepsilon_2\), \(\Delta\), \(\Delta_1\) and \(\Delta_2\) values ​​are calculated, respectively.

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

\(\xi_1 = \displaystyle\frac{\displaystyle \delta }{\displaystyle 3\displaystyle \sqrt[3]{\displaystyle 2}} + \displaystyle\frac{\displaystyle \sqrt[3]{\displaystyle 2}\displaystyle \beta}{3\displaystyle \delta}\)

\(\xi_2 = \displaystyle\frac{B^{2}}{4} -\displaystyle\frac{2C}{3}\)

\(\displaystyle\varepsilon_1 = \displaystyle\frac{-B^{\,3} + 4 B C - 8 D}{4 \displaystyle \sqrt{\displaystyle\xi_1 + \displaystyle\xi_2}}\)

\(\displaystyle\varepsilon_2 = \displaystyle\frac{-\displaystyle \delta}{3 \displaystyle \sqrt[3]{2}} - \displaystyle\frac{\sqrt[3]{2} \beta}{3 \delta} + \displaystyle\frac{B^{\,2} }{2}\)

\(\displaystyle\Delta = \displaystyle\frac{1}{2}\sqrt{\xi_1 + \xi_2}\)
\(\displaystyle\Delta_1 = \displaystyle\frac{1}{2} \sqrt{\displaystyle\varepsilon_2 - \displaystyle\varepsilon_1 - \displaystyle\frac{4 C}{3}}\)
\(\displaystyle\Delta_2 = \displaystyle\frac{1}{2} \sqrt{\displaystyle\varepsilon_2 + \displaystyle\varepsilon_1 - \displaystyle\frac{4 C}{3}}\)

Roots of the equation
Root 1 :       \(\displaystyle\varkappa_1= -\displaystyle\displaystyle\Delta - \displaystyle\Delta_1 - \displaystyle\frac{B}{4}\)

Root 2 :       \(\displaystyle\varkappa_2= -\displaystyle\Delta + \displaystyle\Delta_1 - \displaystyle\frac{B}{4}\)

Root 3 :       \(\displaystyle\varkappa_3= \displaystyle\Delta - \displaystyle\Delta_2 - \displaystyle\frac{B}{4}\)

Root 4 :       \(\displaystyle\varkappa_4= \displaystyle\Delta + \displaystyle\Delta_2 -\displaystyle \frac{B}{4}\)