突然ですが、方程式の解の公式覚えてますか？

2022年3月8日

中学校で二次方程式の解の公式は習うようですが、これは平方完成（省略）を導けば求められます。

二次方程式

$ax^{2}+bx+c=0\quad (a\neq 0)$

の解は

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

ところで、3次方程式の解の公式はあまり教えてくれていません。

これは、カルダノの公式というようで、途中で立方完成を導いて求めるということです。

3次方程式

$a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\quad (a_{3}\neq 0)$

の解の公式は以下の通りであるとWikipediaにあります。

${\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}}$