三次方程式
x 1 = − a 2 3 a 3 + − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 + 4 ( 3 a 1 a 3 − a 2 2 ) 3 + ( − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 ) 2 3 3 2 3 a 3 − 2 3 ( 3 a 1 a 3 − a 2 2 ) 3 a 3 − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 + 4 ( 3 a 1 a 3 − a 2 2 ) 3 + ( − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 ) 2 3 x 2 = − a 2 3 a 3 − ( 1 − i 3 ) − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 + 4 ( 3 a 1 a 3 − a 2 2 ) 3 + ( − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 ) 2 3 6 2 3 a 3 + ( 1 + i 3 ) ( 3 a 1 a 3 − a 2 2 ) 3 × 2 2 3 a 3 − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 + 4 ( 3 a 1 a 3 − a 2 2 ) 3 + ( − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 ) 2 3 x 3 = − a 2 3 a 3 − ( 1 + i 3 ) − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 + 4 ( 3 a 1 a 3 − a 2 2 ) 3 + ( − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 ) 2 3 6 2 3 a 3 + ( 1 − i 3 ) ( 3 a 1 a 3 − a 2 2 ) 3 × 2 2 3 a 3 − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 + 4 ( 3 a 1 a 3 − a 2 2 ) 3 + ( − 2 a 2 3 + 9 a 1 a 3 a 2 − 27 a 0 a 3 2 ) 2 3 {\displaystyle {\begin{aligned}x_{1}=&\color {Red}{-{\frac {a_{2}}{3a_{3}}}}\\&+{\frac {\color {Blue}{\sqrt[{3}]{-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}+{\sqrt {4\left(3a_{1}a_{3}-{a_{2}}^{2}\right)^{3}+\left(-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}\right)^{2}}}}}}{3\color {RedOrange}{{\sqrt[{3}]{2}}a_{3}}}}\\&-{\frac {{\sqrt[{3}]{2}}\color {Green}{\left(3a_{1}a_{3}-{a_{2}}^{2}\right)}}{3a_{3}\color {Blue}{\sqrt[{3}]{-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}+{\sqrt {4\left(3a_{1}a_{3}-{a_{2}}^{2}\right)^{3}+\left(-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}\right)^{2}}}}}}}\\\\x_{2}=&\color {Red}{-{\frac {a_{2}}{3a_{3}}}}\\&-{\frac {\color {RedViolet}{\left(1-i{\sqrt {3}}\right)}{\color {Blue}{\sqrt[{3}]{-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}+{\sqrt {4\left(3a_{1}a_{3}-{a_{2}}^{2}\right)^{3}+\left(-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}\right)^{2}}}}}}}{6\color {RedOrange}{{\sqrt[{3}]{2}}a_{3}}}}\\&+{\frac {\color {MidnightBlue}{\left(1+i{\sqrt {3}}\right)}{\color {Green}{\left(3a_{1}a_{3}-{a_{2}}^{2}\right)}}}{3\times 2^{\tfrac {2}{3}}a_{3}\color {Blue}{\sqrt[{3}]{-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}+{\sqrt {4\left(3a_{1}a_{3}-{a_{2}}^{2}\right)^{3}+\left(-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}\right)^{2}}}}}}}\\\\x_{3}=&\color {Red}{-{\frac {a_{2}}{3a_{3}}}}\\&-{\frac {\color {MidnightBlue}{\left(1+i{\sqrt {3}}\right)}{\color {Blue}{\sqrt[{3}]{-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}+{\sqrt {4\left(3a_{1}a_{3}-{a_{2}}^{2}\right)^{3}+\left(-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}\right)^{2}}}}}}}{6\color {RedOrange}{{\sqrt[{3}]{2}}a_{3}}}}\\&+{\frac {\color {RedViolet}{\left(1-i{\sqrt {3}}\right)}\color {Green}{\left(3a_{1}a_{3}-{a_{2}}^{2}\right)}}{3\times 2^{\tfrac {2}{3}}a_{3}\color {Blue}{\sqrt[{3}]{-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}+{\sqrt {4\left(3a_{1}a_{3}-{a_{2}}^{2}\right)^{3}+\left(-2{a_{2}}^{3}+9a_{1}a_{3}a_{2}-27a_{0}{a_{3}}^{2}\right)^{2}}}}}}}\end{aligned}}}