User:Lucas Brown/Sandbox

General formula of roots

For the general cubic equation $ax^{3}+bx^{2}+cx+d=0$, the general formulas for the roots, in terms of the coefficients, are as follows:

${\begin{aligned}x_{2}=&-{\frac {b}{3a}}\\&+{\frac {1-i{\sqrt {3}}}{6a}}{\sqrt[{3}]{\frac {2b^{3}-9abc+27a^{2}d+{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}}{2}}}\\&+{\frac {1+i{\sqrt {3}}}{6a}}{\sqrt[{3}]{\frac {2b^{3}-9abc+27a^{2}d-{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}}{2}}}\end{aligned}}$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double subscripts: use braces to clarify"): {\displaystyle {\begin{aligned}\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{aligned}}}