Calculate cubic equation

This function returns all three complex roots of the cubic equation:

\(\displaystyle ax^3 + bx^2 + cx + d = 0 \)

To calculate, enter the values for \(a, b, c\) and \(d\), then click on the 'Calculate' button.

Formula for the cubic equation

\(\displaystyle ax^3 + bx^2 + cx + d = 0 \)

Solution according to the Cardano formula

\(\displaystyle x_1=S+T-\frac{b}{3\cdot a}\)

\(\displaystyle x_2=\frac{S+T}{2} +(S-T)i \cdot \frac{\sqrt{3}}{2} -\frac{b}{3\cdot a}\)

\(\displaystyle x_3=-\frac{S+T}{2} -(S-T)i \cdot \frac{\sqrt{3}}{2} -\frac{b}{3\cdot a}\)

\(\displaystyle S=\sqrt[3]{R+\sqrt{Q^3+R^2}} \)

\(\displaystyle T=\sqrt[3]{R-\sqrt{Q^3+R^2}} \)

\(\displaystyle Q=\frac{3 a c-b^2}{9 a^2}\)

\(\displaystyle R=\frac{9 a b c-27 a^2 d-2 b^3}{54 a^3} \)

Mathematical Equations