四次方程
其它方法 > 求根公式

四次方程的求根公式可以通过上述的伽罗瓦理论和因式分解得到。对于 {ax^{4}+bx^{3}+cx^{2}+dx+e=0,a\neq 0}，有：
{x_{1}=-{\frac {b}{4a}}+{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}}}-{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}+{\frac {-b^{3}+4abc-8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}}}}}}}}
{x_{2}=-{\frac {b}{4a}}+{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}}}+{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}-{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}}}}}}}}
{x_{3}=-{\frac {b}{4a}}-{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}}}-{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}+{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}}}}}}}}
{x_{4}=-{\frac {b}{4a}}-{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}}}+{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}+{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {{\sqrt[{3}]{2}}(c^{2}-3bd+12ae)}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}}}}}}}}
{\Delta =256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}}[來源請求]
PlanetMath指出，这四个形式直接使用，即使是在计算机上也过于复杂。这四个解的推导过程的最后几步有较为简单的中间形式可以采用。得到这些解需要用到三次方程的求根公式。