四次方程式
と変形する。ここで上式を展開し係数を比較すると、u の三次方程式
u (p + u) − 4 r u = q
が得られる。このような補助的な方程式を、与えられた四次方程式に関する三次分解方程式(resolvent cubic equation) という。q ≠ 0 なので、この分解方程式の解は u ≠ 0 を満たしており、この解の一つを u として取る。また、求める四次方程式は
{ ( y 2 + p + u 2 ) + u ( y − q 2 u ) } { ( y 2 + p + u 2 ) − u ( y − q 2 u ) } = 0 {\displaystyle \left\{\left(y^{2}+{\frac {p+u}{2}}\right)+{\sqrt {u}}\left(y-{\frac {q}{2u}}\right)\right\}\left\{\left(y^{2}+{\frac {p+u}{2}}\right)-{\sqrt {u}}\left(y-{\frac {q}{2u}}\right)\right\}=0}
となり、この2つの二次方程式から、四次方程式の解を求めることができる。
x 2 = − a 3 4 a 4 − 1 2 a 3 2 4 a 4 2 + 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 + ( 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 ) 2 − 4 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 3 3 2 3 a 4 + 2 3 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 a 4 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 + ( 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 ) 2 − 4 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 3 − 2 a 2 3 a 4 + 1 2 a 3 2 2 a 4 2 − 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 + ( 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 ) 2 − 4 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 3 3 2 3 a 4 − 2 3 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 a 4 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 + ( 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 ) 2 − 4 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 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 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 + ( 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 ) 2 − 4 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 3 3 2 3 a 4 + 2 3 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 a 4 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 + ( 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 ) 2 − 4 ( a 2 2 − 3 a 1 a 3 + 12 a 0 a 4 ) 3 3 − 2 a 2 3 a 4 − 4 a 2 3 a 4 {\displaystyle {\begin{aligned}x_{2}=&\color {Red}{-{\frac {a_{3}}{4a_{4}}}}\\&-{\frac {1}{2}}{\sqrt {\color {Blue}{{\frac {a_{3}^{2}}{4a_{4}^{2}}}+{\frac {\sqrt[{3}]{2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}+{\sqrt {\left(2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}\right){}^{2}-4\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right){}^{3}}}}}{3{\sqrt[{3}]{2}}a_{4}}}+{\frac {{\sqrt[{3}]{2}}\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right)}{3a_{4}{\sqrt[{3}]{2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}+{\sqrt {\left(2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}\right){}^{2}-4\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right){}^{3}}}}}}}-{\frac {2a_{2}}{3a_{4}}}}}}\\&+{\frac {1}{2}}{\sqrt {\color {Green}{{\frac {a_{3}^{2}}{2a_{4}^{2}}}-{\frac {\sqrt[{3}]{2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}+{\sqrt {\left(2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}\right){}^{2}-4\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right){}^{3}}}}}{3{\sqrt[{3}]{2}}a_{4}}}-{\frac {{\sqrt[{3}]{2}}\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right)}{3a_{4}{\sqrt[{3}]{2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}+{\sqrt {\left(2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}\right){}^{2}-4\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right){}^{3}}}}}}}-{\frac {-{\frac {a_{3}^{3}}{a_{4}^{3}}}+{\frac {4a_{2}a_{3}}{a_{4}^{2}}}-{\frac {8a_{1}}{a_{4}}}}{4{\sqrt {{\frac {a_{3}^{2}}{4a_{4}^{2}}}+{\frac {\sqrt[{3}]{2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}+{\sqrt {\left(2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}\right){}^{2}-4\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right){}^{3}}}}}{3{\sqrt[{3}]{2}}a_{4}}}+{\frac {{\sqrt[{3}]{2}}\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right)}{3a_{4}{\sqrt[{3}]{2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}+{\sqrt {\left(2a_{2}^{3}-9a_{1}a_{3}a_{2}-72a_{0}a_{4}a_{2}+27a_{0}a_{3}^{2}+27a_{1}^{2}a_{4}\right){}^{2}-4\left(a_{2}^{2}-3a_{1}a_{3}+12a_{0}a_{4}\right){}^{3}}}}}}}-{\frac {2a_{2}}{3a_{4}}}}}}}-{\frac {4a_{2}}{3a_{4}}}}}}\\\\\end{aligned}}}