\(\small{a*x^4 + b*x^3 + c*x^2 + d*x + e = 0}\) type fourth-degree equations with real or complex coefficients.
This tool returns all roots of the equation as real or complex numbers.
Note 2: If a coefficient is real, the real part must be written in the first box and the second box must be zero.
If the coefficient is complex, write the real part in the first box and the imaginary part in the second box.
By performing the following steps in order, all roots of the fourth-degree equation are obtained.
The coefficients of the equation are divided by \(a\).
The values \(B=\frac{b}{a}\), \(C=\frac{c}{a}\), \(D=\frac{d}{a}\), \(E=\frac{e}{a}\) and the following \(\alpha\) and \(\beta\) are computed.
\(\alpha= 27 E B^{\,2} - 9 B C D + 2 C^{\,3} - 72 E C + 27 D^{\,2} \)
\(\beta=-3 B D + C^2 + 12 E\)
Then, in order, the values \(\delta\), \(\xi_1\), \(\xi_2\), \(\varepsilon_1\), \(\varepsilon_2\), \(\Delta\), \(\Delta_1\), and \(\Delta_2\) are calculated.
\(\delta =\sqrt[3]{\sqrt{\alpha^{\,2} - 4\beta^{\,3}} + \alpha}\)
\(\xi_1 = \frac{\delta}{3\sqrt[3]{2}} + \frac{\sqrt[3]{2}\beta}{3\delta}\)
\(\xi_2 = \frac{B^{2}}{4} - \frac{2C}{3}\)
\(\varepsilon_1 = \frac{-B^{\,3} + 4BC - 8D}{4\sqrt{\xi_1 + \xi_2}}\)
\(\varepsilon_2 = \frac{-\delta}{3\sqrt[3]{2}} - \frac{\sqrt[3]{2}\beta}{3\delta} + \frac{B^{\,2}}{2}\)
\(\Delta = \frac{1}{2}\sqrt{\xi_1 + \xi_2}\)
\(\Delta_1 = \frac{1}{2} \sqrt{\varepsilon_2 - \varepsilon_1 - \frac{4C}{3}}\)
\(\Delta_2 = \frac{1}{2} \sqrt{\varepsilon_2 + \varepsilon_1 - \frac{4C}{3}}\)
Root 1: \(\varkappa_1= -\Delta - \Delta_1 - \frac{B}{4}\)
Root 2: \(\varkappa_2= -\Delta + \Delta_1 - \frac{B}{4}\)
Root 3: \(\varkappa_3= \Delta - \Delta_2 - \frac{B}{4}\)
Root 4: \(\varkappa_4= \Delta + \Delta_2 - \frac{B}{4}\)