Kenan kılıçaslan

  • Chimney Calculation
  • Friction Loss
  • Equation Solve
  • Differential Equation
    Calculations Equations

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 \(\delta\), \(\xi_1\), \(\xi_2\), \(\xi_3\), \(\Delta_1\) 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 :       \(\varkappa_4= \Delta_1 + \Delta_2 -\displaystyle \frac{B}{4}\)

Products    Equation Solver    Calculator    Unit Conversion    Reference    Contact

Pipe Calculations    Chimney Calculation    Air Ducts    Air Conditioning   

Kenan KILIÇASLAN 2012© Copyright.       Designed by Nuit