Equations

# 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.

 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 $$\delta$$, $$\xi_1$$, $$\xi_2$$, $$\xi_3$$, $$\Delta_1$$ 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 :       $$\varkappa_4= \Delta_1 + \Delta_2 -\displaystyle \frac{B}{4}$$
 Equation Solution Nonlinear Equation System Roots Linear Equation System Solution Cubic Equation Solution Quartic Equation Solution Quintic Equation Solution Sextic Equation Solution Differential Equations

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