Quartic equation roots
Returns all the roots of a quartic equation with a real or complex number coefficient.
\(\small{a*x^4+b*x^3+c*x^2+d*x+e=0}\)
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.
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 \(\delta\), \(\xi_1\), \(\xi_2\), \(\xi_3\), \(\Delta_1\) and \(\Delta_2\) values are calculated, respectively.
\(\delta= \displaystyle \sqrt[3]{\sqrt{\alpha^2 - 4 * \beta^3} + \alpha}\)
\(\xi_1= \displaystyle \frac{\delta}{3\sqrt[3]{2}} + \frac{\sqrt[3]{2} * \beta}{3 \delta}\)
\(\xi_2= \displaystyle \frac{B^2}{4} - \frac{2C}{3}\)
\(\xi_3= -B^3 + 4 B C - 8 D\)
\(\Delta_1= \frac{1}{2} \displaystyle \sqrt{\xi_1 +\xi_2}\)
\(\Delta_2= \displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{B^2}{2}- \frac{4C}{3}-\xi_1 -\frac{\xi_3}{4 \sqrt{\xi_1 + \xi_2}} } \)
Roots of the equation
Root 1 : \(\varkappa_1= -\Delta_1 - \Delta_2 - \displaystyle\frac{B}{4}\)
Root 2 : \(\varkappa_2= -\Delta_1 + \Delta_2 - \displaystyle\frac{B}{4}\)
Root 3 : \(\varkappa_3= \Delta_1 - \Delta_2 - \displaystyle\frac{B}{4}\)
Root 4 : \(\varkappa_4= \Delta_1 + \Delta_2 -\displaystyle \frac{B}{4}\)
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