四次方程式
であることが分かる。
この方程式と、3 次の項の消えた四次方程式
x + p x + q x + r = 0
の係数を比べ、p, q, r から a, b, c を求めることができれば、3 次の項の消えた四次方程式の解は上にあるように 4 つ求まる。
実際に係数を比べてみれば
p = − 2 (a + b + c)
q = 8 a b c
r = a + b + c − 2 (a b + b c + c a) = (a + b + c) − 4 (a b + b c + c a)
ここで f0 = (2a), f1 = (2b), f2 = (2c) とおけば
f0 + f1 + f2 = −2p
f0 f1 + f1 f2 + f2 f0 = p − 4 r
f0 f1 f2 = q
となり、根と係数の関係により f0, f1, f2 は三次方程式
u + 2 p u + (p − 4 r) u − q = 0
の解であり、これもフェラーリの方法に現れた三次分解方程式である。この三次方程式を解くことによって a, b, c が得られる。
x 4 = − 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_{4}=&\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}}}