四次方程式
&+\frac{1}{2} \sqrt{\color{Green}{\frac{a_3^2}{2 a_4^2}-\frac{\sqrt[3]{2 a_2^3-9 a_1 a_3 a_2-72 a_0 a_4 a_2+27 a_0 a_3^2+27 a_1^2 a_4+\sqrt{\left(2 a_2^3-9 a_1 a_3 a_2-72 a_0 a_4 a_2+27 a_0 a_3^2+27 a_1^2 a_4\right){}^2-4 \left(a_2^2-3 a_1 a_3+12 a_0 a_4\right){}^3}}{3 \sqrt[3]{2} a_4}-\frac{\sqrt[3]{2} \left(a_2^2-3 a_1 a_3+12 a_0 a_4\right)}{3 a_4 \sqrt[3]{2 a_2^3-9 a_1 a_3 a_2-72 a_0 a_4 a_2+27 a_0 a_3^2+27 a_1^2 a_4+\sqrt{\left(2 a_2^3-9 a_1 a_3 a_2-72 a_0 a_4 a_2+27 a_0 a_3^2+27 a_1^2 a_4\right){}^2-4 \left(a_2^2-3 a_1 a_3+12 a_0 a_4\right){}^3}}+\frac{-\frac{a_3^3}{a_4^3}+\frac{4 a_2 a_3}{a_4^2}-\frac{8 a_1}{a_4}{4 \sqrt{\frac{a_3^2}{4 a_4^2}+\frac{\sqrt[3]{2 a_2^3-9 a_1 a_3 a_2-72 a_0 a_4 a_2+27 a_0 a_3^2+27 a_1^2 a_4+\sqrt{\left(2 a_2^3-9 a_1 a_3 a_2-72 a_0 a_4 a_2+27 a_0 a_3^2+27 a_1^2 a_4\right){}^2-4 \left(a_2^2-3 a_1 a_3+12 a_0 a_4\right){}^3}}{3 \sqrt[3]{2} a_4}+\frac{\sqrt[3]{2} \left(a_2^2-3 a_1 a_3+12 a_0 a_4\right)}{3 a_4 \sqrt[3]{2 a_2^3-9 a_1 a_3 a_2-72 a_0 a_4 a_2+27 a_0 a_3^2+27 a_1^2 a_4+\sqrt{\left(2 a_2^3-9 a_1 a_3 a_2-72 a_0 a_4 a_2+27 a_0 a_3^2+27 a_1^2 a_4\right){}^2-4 \left(a_2^2-3 a_1 a_3+12 a_0 a_4\right){}^3}}-\frac{2 a_2}{3 a_4}}-\frac{4 a_2}{3 a_4}} \\
\\
\end{align}
\begin{align}
& a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0 \quad (a_4\ne 0) \\
\\
& x_1 = {\color{Red}{-\frac{a_3}{4a_4}} - \frac{\sqrt\color{Blue}{\begin{array} \hline 2 \\ \hline \end{array}}{2} - \frac{\sqrt\color{Green}{\begin{array} \hline 4 \\ \hline \end{array}}{2} \\
& x_2 = {\color{Red}{-\frac{a_3}{4a_4}} + \frac{\sqrt\color{Blue}{\begin{array} \hline 2 \\ \hline \end{array}}{2} - \frac{\sqrt\color{Green}{\begin{array} \hline 4 \\ \hline \end{array}}{2} \\
& x_3 = {\color{Red}{-\frac{a_3}{4a_4}} - \frac{\sqrt\color{Blue}{\begin{array} \hline 1 \\ \hline \end{array}}{2} + \frac{\sqrt\color{Green}{\begin{array} \hline 4 \\ \hline \end{array}}{2} \\
& x_4 = {\color{Red}{-\frac{a_3}{4a_4}} + \frac{\sqrt\color{Blue}{\begin{array} \hline 1 \\ \hline \end{array}}{2} + \frac{\sqrt\color{Green}{\begin{array} \hline 4 \\ \hline \end{array}}{2} \\
\\
& \begin{array}
\hline \begin{array} \hline 1 \\ \hline \end{array} \rightarrow \frac{\begin{array} \hline 3 \\ \hline \end{array}{4\sqrt{\begin{array} \hline 4 \\ \hline \end{array}} + \frac{a_3^2}{2a_4^2} - \frac{2^{\frac{1}{3} \begin{array} \hline 5 \\ \hline \end{array}{3\begin{array} \hline 8 \\ \hline \end{array}^{\frac{1}{3}a_4} - \frac{\begin{array} \hline 8 \\ \hline \end{array}^{\frac{1}{3}}{3 \times 2^{\frac{1}{3}a_4} - \frac{4a_2}{3a_4} \\
\hline \begin{array} \hline 2 \\ \hline \end{array} \rightarrow -\frac{\begin{array} \hline 3 \\ \hline \end{array}{4\sqrt{\begin{array} \hline 4 \\ \hline \end{array}} + \frac{a_3^2}{2a_4^2} - \frac{2^{\frac{1}{3} \begin{array} \hline 5 \\ \hline \end{array}{3\begin{array} \hline 8 \\ \hline \end{array}^{\frac{1}{3}a_4} - \frac{\begin{array} \hline 8 \\ \hline \end{array}^{\frac{1}{3}}{3 \times 2^{\frac{1}{3}a_4} - \frac{4a_2}{3a_4} \\
\hline \begin{array} \hline 3 \\ \hline \end{array} \rightarrow - \frac{a_3^3}{a_4^3} + \frac{4a_2a_3}{a_4^2} - \frac{8a_1}{a_4} \\
\hline \begin{array} \hline 4 \\ \hline \end{array} \rightarrow \frac{a_3^2}{4a_4^2} + \frac{2^{\frac{1}{3} \begin{array} \hline 5 \\ \hline \end{array}{3\begin{array} \hline 8 \\ \hline \end{array}^{\frac{1}{3}a_4} + \frac{\begin{array} \hline 8 \\ \hline \end{array}^{\frac{1}{3}}{3 \times 2^{\frac{1}{3}a_4} - \frac{2a_2}{3a_4} \\
\hline \begin{array} \hline 5 \\ \hline \end{array} \rightarrow a_2^2 - 3a_1a_3 + 12a_0a_4 \\
\hline \begin{array} \hline 6 \\ \hline \end{array} \rightarrow 2a_2^3 - 9a_1a_2a_3 + 27a_0a_3^2 + 27 a_1^2a_4 - 72a_0a_2a_4 \\
\hline \begin{array} \hline 7 \\ \hline \end{array} \rightarrow - 4\begin{array} \hline 5 \\ \hline \end{array}^3 + \begin{array} \hline 6 \\ \hline \end{array}^2 \\
\hline \begin{array} \hline 8 \\ \hline \end{array} \rightarrow \begin{array} \hline 6 \\ \hline \end{array} + \sqrt{\begin{array} \hline 7 \\ \hline \end{array} \\
\hline
\end{array}
\end{align}
地球を扁球回転楕円体とみなしたとき、地心直交座標 (x,y,z) から地理座標(緯度 \varphi、経度 \lambda、高度(楕円体高)h)への座標換算を行う際に四次方程式が現れる。経緯度#地理経緯度の変換式
(代数的な解の公式が存在しない)五次方程式の解をレベル5のモジュラー方程式の解を利用する方法により得ようとするプロセスの途中で四次方程式が現れる。