User:Lucas Brown/Sandbox

General formula of roots
For the general cubic equation $$a x^3 + b x^2 + c x + d = 0$$, the general formulas for the roots, in terms of the coefficients, are as follows:


 * $$\begin{align}

x_2 = &-\frac{b}{3 a}\\ &+\frac{1-i \sqrt{3}}{6 a} \sqrt[3]{\frac{2 b^3-9 a b c+27 a^2 d+\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}}{2}}\\ &+\frac{1+i \sqrt{3}}{6 a} \sqrt[3]{\frac{2 b^3-9 a b c+27 a^2 d-\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}}{2}} \end{align}$$


 * $$\begin{align}

\langle \ddot{r}_1_x, \ddot{r}_1_y, \ddot{r}_1_z \rangle = &\frac{\delta \left( \langle r_1_x, r_1_y, r_1_z \rangle - \langle r_1_y, r_1_x, r_1_z \rangle \right) + \epsilon \langle \dot{r}_1_x, \dot{r}_1_y, \dot{r}_1_z \rangle \times \left(\left(\langle \dot{r}_1_y, \dot{r}_1_x, \dot{r}_1_z \rangle \times \left(\langle r_1_x, r_1_y, r_1_z \rangle - \langle r_1_y, r_1_x, r_1_z \rangle \right) \right) \right)}{m_e \mid \mid \langle r_1_x, r_1_y, r_1_z \rangle - \langle r_1_y, r_1_x, r_1_z \rangle \mid \mid ^3} \\ \langle \ddot{r}_1_y, \ddot{r}_1_x, \ddot{r}_1_z \rangle = &\frac{\delta \left( \langle r_1_y, r_1_x, r_1_z \rangle - \langle r_1_x, r_1_y, r_1_z \rangle \right) + \epsilon \langle \dot{r}_1_y, \dot{r}_1_x, \dot{r}_1_z \rangle \times \left(\left(\langle \dot{r}_1_x, \dot{r}_1_y, \dot{r}_1_z \rangle \times \left(\langle r_1_y, r_1_x, r_1_z \rangle - \langle r_1_x, r_1_y, r_1_z \rangle \right) \right) \right)}{m_e \mid \mid \langle r_1_x, r_1_y, r_1_z \rangle - \langle r_1_y, r_1_x, r_1_z \rangle \mid \mid ^3} \end{align}$$