四次方程式
a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = 0 ( a 4 ≠ 0 ) x 1 = − a 3 4 a 4 − 2 2 − 4 2 x 2 = − a 3 4 a 4 + 2 2 − 4 2 x 3 = − a 3 4 a 4 − 1 2 + 4 2 x 4 = − a 3 4 a 4 + 1 2 + 4 2 1 → 3 4 4 + a 3 2 2 a 4 2 − 2 1 3 5 3 8 1 3 a 4 − 8 1 3 3 × 2 1 3 a 4 − 4 a 2 3 a 4 2 → − 3 4 4 + a 3 2 2 a 4 2 − 2 1 3 5 3 8 1 3 a 4 − 8 1 3 3 × 2 1 3 a 4 − 4 a 2 3 a 4 3 → − a 3 3 a 4 3 + 4 a 2 a 3 a 4 2 − 8 a 1 a 4 4 → a 3 2 4 a 4 2 + 2 1 3 5 3 8 1 3 a 4 + 8 1 3 3 × 2 1 3 a 4 − 2 a 2 3 a 4 5 → a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 6 → 2 a 2 3 − 9 a 1 a 2 a 3 + 27 a 0 a 3 2 + 27 a 1 2 a 4 − 72 a 0 a 2 a 4 7 → − 4 5 3 + 6 2 8 → 6 + 7 {\displaystyle {\begin{aligned}&a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\quad (a_{4}\neq 0)\\\\&x_{1}={\color {Red}{-{\frac {a_{3}}{4a_{4}}}}}-{\frac {\sqrt {\color {Blue}{\begin{array}{|c|}\hline 2\\\hline \end{array}}}}{2}}-{\frac {\sqrt {\color {Green}{\begin{array}{|c|}\hline 4\\\hline \end{array}}}}{2}}\\&x_{2}={\color {Red}{-{\frac {a_{3}}{4a_{4}}}}}+{\frac {\sqrt {\color {Blue}{\begin{array}{|c|}\hline 2\\\hline \end{array}}}}{2}}-{\frac {\sqrt {\color {Green}{\begin{array}{|c|}\hline 4\\\hline \end{array}}}}{2}}\\&x_{3}={\color {Red}{-{\frac {a_{3}}{4a_{4}}}}}-{\frac {\sqrt {\color {Blue}{\begin{array}{|c|}\hline 1\\\hline \end{array}}}}{2}}+{\frac {\sqrt {\color {Green}{\begin{array}{|c|}\hline 4\\\hline \end{array}}}}{2}}\\&x_{4}={\color {Red}{-{\frac {a_{3}}{4a_{4}}}}}+{\frac {\sqrt {\color {Blue}{\begin{array}{|c|}\hline 1\\\hline \end{array}}}}{2}}+{\frac {\sqrt {\color {Green}{\begin{array}{|c|}\hline 4\\\hline \end{array}}}}{2}}\\\\&{\begin{array}{|l|}\hline {\begin{array}{|c|}\hline 1\\\hline \end{array}}\rightarrow {\frac {\begin{array}{|c|}\hline 3\\\hline \end{array}}{4{\sqrt {\begin{array}{|c|}\hline 4\\\hline \end{array}}}}}+{\frac {a_{3}^{2}}{2a_{4}^{2}}}-{\frac {2^{\frac {1}{3}}{\begin{array}{|c|}\hline 5\\\hline \end{array}}}{3{\begin{array}{|c|}\hline 8\\\hline \end{array}}^{\frac {1}{3}}a_{4}}}-{\frac {{\begin{array}{|c|}\hline 8\\\hline \end{array}}^{\frac {1}{3}}}{3\times 2^{\frac {1}{3}}a_{4}}}-{\frac {4a_{2}}{3a_{4}}}\\\hline {\begin{array}{|c|}\hline 2\\\hline \end{array}}\rightarrow -{\frac {\begin{array}{|c|}\hline 3\\\hline \end{array}}{4{\sqrt {\begin{array}{|c|}\hline 4\\\hline \end{array}}}}}+{\frac {a_{3}^{2}}{2a_{4}^{2}}}-{\frac {2^{\frac {1}{3}}{\begin{array}{|c|}\hline 5\\\hline \end{array}}}{3{\begin{array}{|c|}\hline 8\\\hline \end{array}}^{\frac {1}{3}}a_{4}}}-{\frac {{\begin{array}{|c|}\hline 8\\\hline \end{array}}^{\frac {1}{3}}}{3\times 2^{\frac {1}{3}}a_{4}}}-{\frac {4a_{2}}{3a_{4}}}\\\hline {\begin{array}{|c|}\hline 3\\\hline \end{array}}\rightarrow -{\frac {a_{3}^{3}}{a_{4}^{3}}}+{\frac {4a_{2}a_{3}}{a_{4}^{2}}}-{\frac {8a_{1}}{a_{4}}}\\\hline {\begin{array}{|c|}\hline 4\\\hline \end{array}}\rightarrow {\frac {a_{3}^{2}}{4a_{4}^{2}}}+{\frac {2^{\frac {1}{3}}{\begin{array}{|c|}\hline 5\\\hline \end{array}}}{3{\begin{array}{|c|}\hline 8\\\hline \end{array}}^{\frac {1}{3}}a_{4}}}+{\frac {{\begin{array}{|c|}\hline 8\\\hline \end{array}}^{\frac {1}{3}}}{3\times 2^{\frac {1}{3}}a_{4}}}-{\frac {2a_{2}}{3a_{4}}}\\\hline {\begin{array}{|c|}\hline 5\\\hline \end{array}}\rightarrow a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\\\hline {\begin{array}{|c|}\hline 6\\\hline \end{array}}\rightarrow 2a_{2}^{3}-9a_{1}a_{2}a_{3}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}-72a_{0}a_{2}a_{4}\\\hline {\begin{array}{|c|}\hline 7\\\hline \end{array}}\rightarrow -4{\begin{array}{|c|}\hline 5\\\hline \end{array}}^{3}+{\begin{array}{|c|}\hline 6\\\hline \end{array}}^{2}\\\hline {\begin{array}{|c|}\hline 8\\\hline \end{array}}\rightarrow {\begin{array}{|c|}\hline 6\\\hline \end{array}}+{\sqrt {\begin{array}{|c|}\hline 7\\\hline \end{array}}}\\\hline \end{array}}\end{aligned}}}