Third-Degree Equation Solver

\(\small{a*x^3+b*x^2+c*x+d=0}\) type cubic equations with real or complex coefficients. The tool returns all roots of the equation as real or complex numbers.

Note 1: \(a \ne 0\) must hold.

Note 2: If a coefficient is a real number, write the real part in the first box and set the second box to zero. If a coefficient is a complex number, write the real part in the first box and the imaginary part in the second box.

From formulas (1) and (2), the values of \(\alpha\) and \(\beta\) are obtained. Using these values, the value of \(\Delta\) is calculated. From \(\alpha\), \(\beta\), and \(\Delta\), the roots \(x_1\), \(x_2\), and \(x_3\) are determined. The symbol \(i\) in the formulas represents the imaginary unit and is equal to \(i=\sqrt{-1}\). \begin{equation} \alpha=\frac{d}{2a} + \frac{b^3}{27a^3} - \frac{bc}{6a^2} \end{equation} \begin{equation} \beta=\frac{c}{3a} -\frac{b^2}{9a^2} \end{equation} \begin{equation} \Delta= \sqrt{\alpha^2 + \beta^3}-\alpha \end{equation} \begin{equation} x_1=\sqrt[3]{\Delta} -\frac{b}{3a} - \frac{\beta}{\sqrt[3]{\Delta}} \end{equation} \begin{equation} \begin{array}{ll} x_2 &=\displaystyle \frac{\beta}{2 \sqrt[3]{\Delta }} -\frac{b}{3a}-\frac{1}{2}\sqrt[3]{\Delta} \\ &-\displaystyle \frac{\sqrt{3}}{2}\left\{\displaystyle \frac{\beta}{\sqrt[3]{\Delta }} + \sqrt[3]{\Delta }\right\} i \end{array} \end{equation} \begin{equation} \begin{array}{ll} x_3 &=\displaystyle \frac{\beta}{2 \sqrt[3]{\Delta}} -\frac{b}{3a}-\frac{1}{2}\sqrt[3]{\Delta } \\ &+\displaystyle \frac{\sqrt{3}}{2}\left\{\displaystyle \frac{\beta}{\sqrt[3]{\Delta}} + \sqrt[3]{\Delta }\right\} i \end{array} \end{equation}