四次方程式
レオンハルト・オイラーは、三次方程式や四次方程式の解法をいくつか発見した。ここに述べる方法もオイラーの方法と呼ばれる解法の一つである。
(x + a + b + c) (x + a − b − c) (x − a + b − c) (x − a − b + c)
= {(x + a) − (b + c)}{(x − a) − (b − c)}
= (x − a) + (b − c) − (x + a) (b − c) − (x − a) (b + c)
= x + a + b + c − 2 (x a + x b + x c + a b + b c + c a) + 8 x a b c
= x − 2 (a + b + c) x + 8 a b c x + a + b + c − 2 (a b + b c + c a)
という等式を用いると、x を未知数とする四次方程式
x − 2 (a + b + c) x + 8 a b c x + a + b + c − 2 (a b + b c + c a) = 0
の4個の解は
− a − b − c, − a + b + c, a − b + c, a + b − c
x 1 = − 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_{1}=&\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}}}
x 3 = − 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_{3}=&\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}}}
四次方程式の内奇数次の項が無い
a4 x + a2 x + a0 = 0 (a4 ≠ 0)
の形の式は x を変数とする二次方程式と見ることができ、複二次方程式 (biquadratic equation)、左辺は複二次式と呼ばれる。二次方程式の解法を知っていれば簡単に解くことができる。
y = x と変換することで y に関する二次方程式
a4 y + a2 y + a0 = 0
を得ることができ、この二次方程式を解くことによって解を求められる。
また、実数を係数とする複二次式
x + A2 x + A0
に対して、次のような二次式の積への因数分解もよく行われる。x の二次方程式とみたときの判別式
D = A2 − 4A0
の符号によって
D > 0 であれば x について平方完成することにより
u = r 0 + i r 1 − r 2 − i r 3 {\displaystyle u=r_{0}+ir_{1}-r_{2}-ir_{3}}
の形の式、さらに一般に、n次方程式であれば 1の原始n乗根 ζ n {\displaystyle \zeta _{n}} を用いて
u = ∑ k = 0 n − 1 ζ n k r k = r 0 + ζ n r 1 + ζ n 2 r 2 + ⋯ + ζ n n − 2 r n − 2 + ζ n n − 1 r n − 1 {\displaystyle u=\sum _{k=0}^{n-1}{\zeta _{n}}^{k}r_{k}=r_{0}+\zeta _{n}r_{1}+{\zeta _{n}}^{2}r_{2}+\cdots +{\zeta _{n}}^{n-2}r_{n-2}+{\zeta _{n}}^{n-1}r_{n-1}}
の形の式の性質を詳しく調べたが、五次以上の代数方程式の代数的解法の発見には至らなかった。この形の式をラグランジュの分解式 (Lagrange resolvent) という。 五次以上の代数方程式の代数的解法の存在については、パオロ・ルフィニ、オーギュスタン=ルイ・コーシー、ニールス・アーベルらの研究がアーベル-ルフィニの定理として結実し、否定されることになるが、彼らの研究は、このようなラグランジュの研究を源流としている。