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【原創(chuàng)】一元四次方程求根公式推導(dǎo)(可能有誤,敬請糾正)

2023-08-11 03:52 作者:中國大黃鴨鴨  | 我要投稿

  任意一元四次方程都可以表示成如下形式:

ax%5E4%2Bbx%5E3%2Bcx%5E2%2Bdx%2Be%3D0(a%E2%89%A00)

  等式兩邊同時除以a(a%E2%89%A00),得:

x%5E4%2B%5Cfrac%7Bb%7D%7Ba%7Dx%5E3%2B%5Cfrac%7Bc%7D%7Ba%7Dx%5E2%3D-%5Cfrac%7Bd%7D%7Ba%7Dx-%5Cfrac%7Be%7D%7Ba%7D(a%E2%89%A00)

  將%5Cfrac%7Bc%7D%7Ba%7Dx%5E2%2B%5Cfrac%7Bd%7D%7Ba%7Dx%2B%5Cfrac%7Be%7D%7Ba%7D移項到等式右邊,得:

x%5E4%2B%5Cfrac%7Bb%7D%7Ba%7Dx%5E3%3D-%5Cfrac%7Bc%7D%7Ba%7Dx%5E2-%5Cfrac%7Bd%7D%7Ba%7Dx-%5Cfrac%7Be%7D%7Ba%7D(a%E2%89%A00)

  從%5Cfrac%7Bb%7D%7Ba%7Dx%5E3中提取因數(shù)2,得:

x%5E4%2B2%5Cfrac%7Bb%7D%7B2a%7Dx%5E3%3D-%5Cfrac%7Bc%7D%7Ba%7Dx%5E2-%5Cfrac%7Bd%7D%7Ba%7Dx-%5Cfrac%7Be%7D%7Ba%7D(a%E2%89%A00)

  等式兩邊同時加上%5Cfrac%7Bb%5E2%7D%7B4a%5E2%7D,得:

x%5E4%2B2%5Cfrac%7Bb%7D%7B2a%7Dx%5E3%2B%5Cfrac%7Bb%5E2%7D%7B4a%5E2%7Dx%5E2%20%3D%20%5Cleft(%20%5Cfrac%7Bb%5E2%7D%7B4a%5E2%7D-%5Cfrac%7Bc%7D%7Ba%7D%5Cright)x%5E2-%5Cfrac%7Bd%7D%7Ba%7Dx-%5Cfrac%7Be%7D%7Ba%7D%20(a%E2%89%A00)

  等式左邊配完全平方:

%5Cleft(%20x%5E2%2B%5Cfrac%7Bb%7D%7B2a%7Dx%20%5Cright)%5E2%20%3D%20%5Cleft(%20%5Cfrac%7Bb%5E2%7D%7B4a%5E2%7D-%5Cfrac%7Bc%7D%7Ba%7D%5Cright)x%5E2-%5Cfrac%7Bd%7D%7Ba%7Dx-%5Cfrac%7Be%7D%7Ba%7D%20(a%E2%89%A00)

  引入常數(shù)%E2%88%86_3,等式兩邊同時加上2%5Cleft(%20x%5E2%2B%5Cfrac%7Bb%7D%7B2a%7D%20%5Cright)%20%5Cfrac%7B%E2%88%86_3%7D%7B2%7D%2B%5Cfrac%7B%E2%88%86_3%5E2%7D%7B4%7D,得:

%5Cleft(%20x%5E2%2B%5Cfrac%7Bb%7D%7B2a%7Dx%20%5Cright)%5E2%20%2B%202%5Cleft(%20x%5E2%2B%5Cfrac%7Bb%7D%7B2a%7Dx%20%5Cright)%20%5Cfrac%7B%E2%88%86_3%7D%7B2%7D%2B%5Cfrac%7B%E2%88%86_3%5E2%7D%7B4%7D%20%3D%20%5Cleft(%20%5Cfrac%7Bb%5E2%7D%7B4a%5E2%7D-%5Cfrac%7Bc%7D%7Ba%7D%5Cright)x%5E2-%5Cfrac%7Bd%7D%7Ba%7Dx%20-%20%5Cfrac%7Be%7D%7Ba%7D%20%2B%202%5Cleft(%20x%5E2%2B%5Cfrac%7Bb%7D%7B2a%7D%20%5Cright)%20%5Cfrac%7B%E2%88%86_3%7D%7B2%7D%2B%5Cfrac%7B%E2%88%86_3%5E2%7D%7B4%7D(a%E2%89%A00)

  等式左邊配完全平方,按關(guān)于x的降冪排序整理得:

%5Cleft(%20x%5E2%2B%5Cfrac%7Bb%7D%7B2a%7Dx%20%2B%20%5Cfrac%7B%E2%88%86_3%7D%7B2%7D%20%5Cright)%5E2%20%3D%20%5Cleft(%20%5Cfrac%7Bb%5E2%7D%7B4a%5E2%7D%20%2B%20%E2%88%86_3%20-%20%5Cfrac%7Bc%7D%7Ba%7D%20%5Cright)%20x%5E2%20%2B%20%5Cleft(%20%5Cfrac%7Bb%7D%7B2a%7D%E2%88%86_3-%5Cfrac%7Bd%7D%7Ba%7D%20%5Cright)x%20%2B%5Cleft(%20%5Cfrac%7B%E2%88%86_3%5E2%7D%7B4%7D%20-%20%5Cfrac%7Be%7D%7Ba%7D%20%5Cright)%20(a%E2%89%A00)

  為使等式右邊配成完全平方,從而兩邊同時開平方形成次數(shù)較低的方程,則右側(cè)關(guān)于x的二次三項式的判別式必須等于0,即:

%5Cleft(%20%5Cfrac%7Bb%7D%7B2a%7D%E2%88%86_3-%5Cfrac%7Bd%7D%7Ba%7D%20%5Cright)%5E2-4%5Cleft(%20%5Cfrac%7Bb%5E2%7D%7B4a%5E2%7D%20%2B%20%E2%88%86_3%20-%20%5Cfrac%7Bc%7D%7Ba%7D%20%5Cright)%5Cleft(%20%5Cfrac%7B%E2%88%86_3%5E2%7D%7B4%7D%20-%20%5Cfrac%7Be%7D%7Ba%7D%20%5Cright)%20%3D%200%20(a%E2%89%A00)

  整理得:

-%E2%88%86_3%5E3%2B%5Cfrac%7Bc%7D%7Ba%7D%E2%88%86_3%5E2%2B%5Cfrac%7B4ae-bd%7D%7Ba%5E2%7D%E2%88%86_3%2B%5Cfrac%7Bb%5E2e%2Bad%5E2-4ace%7D%7Ba%5E3%7D%3D0%20(a%E2%89%A00)

  由卡爾丹公式

Ay%5E3%2BBy%5E2%2BCy%2Bd%3D0(A%E2%89%A00)

P%3D%5Cfrac%7B3AC-B%5E2%7D%7B3A%5E2%7D%2CQ%3D%5Cfrac%7B2B%5E3%2B27A%5E2%20D-9ABC%7D%7B27A%5E3%7D(A%E2%89%A00)

y%3D%5Csqrt%5B3%5D%7B-%5Cfrac%7BQ%7D%7B2%7D%2B%5Csqrt%7B%5Cfrac%7BQ%5E2%7D%7B4%7D%2B%5Cfrac%7BP%5E3%7D%7B27%7D%7D%7D%2B%5Csqrt%5B3%5D%7B-%5Cfrac%7BQ%7D%7B2%7D-%5Csqrt%7B%5Cfrac%7BQ%5E2%7D%7B4%7D%2B%5Cfrac%7BP%5E3%7D%7B27%7D%7D%7D%20

  得:

A%3D-1%2CB%3D%5Cfrac%7Bc%7D%7Ba%7D%2CC%3D%5Cfrac%7B4ae-bd%7D%7Ba%5E2%7D%2CD%3D%5Cfrac%7Bb%5E2e%2Bad%5E2-4ace%7D%7Ba%5E3%7D


  P%3D%5Cfrac%7B3AC-B%5E2%7D%7B3A%5E2%7D

   ?%3D%5Cfrac%7B-3%5Cfrac%7B4ae-bd%7D%7Ba%5E2%7D-%5Cfrac%7Bc%5E2%7D%7Ba%5E2%7D%7D%7B3%7D

   ?%3D-%5Cfrac%7Bc%5E2-3bd-12ae%7D%7B3a%5E2%7D


  Q%3D%5Cfrac%7B2B%5E3%2B27A%5E2%20D-9ABC%7D%7B27A%5E3%7D

   ?%3D-%5Cfrac%7B2%5Cfrac%7Bc%5E3%7D%7Ba%5E3%7D%2B27%5Cfrac%7Bb%5E2e%2Bad%5E2-4ace%7D%7Ba%5E3%7D%2B9%5Cfrac%7B4ace-bcd%7D%7Ba%5E3%7D%7D%7B27%7D

   ?%3D-%5Cfrac%7B2c%5E3-9bcd%2B27ad%5E2%2B27b%5E2e-72ace%7D%7B27a%5E3%7D


  令%E2%88%86_1%3Dc%5E2-3bd-12ae,%E2%88%86_2%3D2c%5E3-9bcd%2B27ad%5E2%2B27b%5E2e-72ace,則:

P%3D-%5Cfrac%7B%E2%88%86_1%7D%7B3a%5E2%7D%2CQ%3D-%5Cfrac%7B%E2%88%86_2%7D%7B27a%5E3%7D

  將P%3D-%5Cfrac%7B%E2%88%86_1%7D%7B3a%5E2%7D%2CQ%3D-%5Cfrac%7B%E2%88%86_2%7D%7B27a%5E3%7D代入卡爾丹公式,得:

  %E2%88%86_3%3D%5Csqrt%5B3%5D%7B-%5Cfrac%7BQ%7D%7B2%7D%2B%5Csqrt%7B%5Cfrac%7BQ%5E2%7D%7B4%7D%2B%5Cfrac%7BP%5E3%7D%7B27%7D%7D%7D%2B%5Csqrt%5B3%5D%7B-%5Cfrac%7BQ%7D%7B2%7D-%5Csqrt%7B%5Cfrac%7BQ%5E2%7D%7B4%7D%2B%5Cfrac%7BP%5E3%7D%7B27%7D%7D%7D%20

    %3D%5Csqrt%5B3%5D%7B%5Cfrac%7B%E2%88%86_2%7D%7B54a%5E3%7D%2B%5Csqrt%7B%5Cfrac%7B%E2%88%86_2%5E2%7D%7B2916a%5E6%7D-%5Cfrac%7B%E2%88%86_1%5E3%7D%7B729a%5E6%7D%7D%7D%2B%5Csqrt%5B3%5D%7B%5Cfrac%7B%E2%88%86_2%7D%7B54a%5E3%7D-%5Csqrt%7B%5Cfrac%7B%E2%88%86_2%5E2%7D%7B2916a%5E6%7D-%5Cfrac%7B%E2%88%86_1%5E3%7D%7B729a%5E6%7D%7D%7D%20

    %3D%5Cfrac%7B%5Csqrt%5B3%5D%7B4%E2%88%86_2%2B4%5Csqrt%7B%E2%88%86_2%5E2-4%E2%88%86_1%5E3%7D%7D%2B%5Csqrt%5B3%5D%7B4%E2%88%86_2-4%5Csqrt%7B%E2%88%86_2%5E2-4%E2%88%86_1%5E3%7D%7D%7D%7B3a%7D%20

  現(xiàn)在看含常數(shù)項%E2%88%86_3的原方程變形式:

%5Cleft(%20x%5E2%2B%5Cfrac%7Bb%7D%7B2a%7Dx%20%2B%20%5Cfrac%7B%E2%88%86_3%7D%7B2%7D%20%5Cright)%5E2%20%3D%20%5Cleft(%20%5Cfrac%7Bb%5E2%7D%7B4a%5E2%7D%20%2B%20%E2%88%86_3%20-%20%5Cfrac%7Bc%7D%7Ba%7D%20%5Cright)%20x%5E2%20%2B%20%5Cleft(%20%5Cfrac%7Bb%7D%7B2a%7D%E2%88%86_3-%5Cfrac%7Bd%7D%7Ba%7D%20%5Cright)x%20%2B%5Cleft(%20%5Cfrac%7B%E2%88%86_3%5E2%7D%7B4%7D%20-%20%5Cfrac%7Be%7D%7Ba%7D%20%5Cright)%20(a%E2%89%A00)

  等式兩邊同時乘4,整理得:

%5Cleft(%202x%5E2%2B%5Cfrac%7Bb%7D%7Ba%7Dx%20%2B%20%E2%88%86_3%20%5Cright)%5E2%20%3D%20%5Cleft(%20%5Cfrac%7Bb%5E2-4ac%7D%7Ba%5E2%7D%20%2B%204%E2%88%86_3%20%5Cright)%20x%5E2%20%2B%20%5Cleft(%20%5Cfrac%7B2b%E2%88%86_3-4d%7D%7Ba%7D%20%5Cright)x%20%2B%5Cleft(%20%E2%88%86_3%5E2%20-%20%5Cfrac%7B4e%7D%7Ba%7D%20%5Cright)%20(a%E2%89%A00)

  令%E2%88%86%3D%5Csqrt%7B%5Cfrac%7Bb%5E2-4ac%7D%7Ba%5E2%7D%20%2B%204%E2%88%86_3%7D%20,則:

%5Cleft(%202x%5E2%2B%5Cfrac%7Bb%7D%7Ba%7Dx%20%2B%20%E2%88%86_3%20%5Cright)%5E2%20%3D%20%E2%88%86%5E2x%5E2%20%2B%20%5Cleft(%20%5Cfrac%7B2b%E2%88%86_3-4d%7D%7Ba%7D%20%5Cright)x%20%2B%5Cleft(%20%E2%88%86_3%5E2%20-%20%5Cfrac%7B4e%7D%7Ba%7D%20%5Cright)%20(a%E2%89%A00)

  由于等式右邊關(guān)于x的二次三項式判別式等于0,則等式右邊可以配成完全平方:

%5Cleft(%202x%5E2%2B%5Cfrac%7Bb%7D%7Ba%7Dx%20%2B%20%E2%88%86_3%20%5Cright)%5E2%20%3D%20%5Cleft(%20%E2%88%86%20x%20%2B%20%5Cfrac%7B%20%5Cfrac%7B2b%E2%88%86_3-4d%7D%7Ba%7D%20%7D%7B2%E2%88%86%7D%20%5Cright)%5E2%20(a%E2%89%A00)

  等式兩邊同時開平方,整理得:

2x%5E2%2B%5Cfrac%7Bb%7D%7Ba%7Dx%20%2B%20%E2%88%86_3%20%3D%20%E2%88%93%5Cleft(%20%E2%88%86%20x%20%2B%20%5Cfrac%7Bb%E2%88%86_3-2d%7D%7Ba%E2%88%86%7D%20%5Cright)%20(a%E2%89%A00)

  移項,得:

2x%5E2%2B%5Cleft(%5Cfrac%7Bb%7D%7Ba%7D%C2%B1%E2%88%86%5Cright)x%20%2B%20%5Cleft(%20%E2%88%86_3%20%C2%B1%20%5Cfrac%7Bb%E2%88%86_3-2d%7D%7Ba%E2%88%86%7D%20%5Cright)%3D%200%20(a%E2%89%A00)

  解得:

x_1%20%3D%20%5Cfrac%7B-%5Cleft(%20%5Cfrac%7Bb%7D%7Ba%7D%20%2B%20%E2%88%86%20%5Cright)%20-%20%5Csqrt%7B%5Cleft(%20%5Cfrac%7Bb%7D%7Ba%7D%20%2B%E2%88%86%20%5Cright)%5E2-8%5Cleft(%20%E2%88%86_3%20%2B%20%5Cfrac%7Bb%E2%88%86_3-2d%7D%7Ba%E2%88%86%7D%20%5Cright)%7D%20%7D%7B4%7D%20

x_2%20%3D%20%5Cfrac%7B-%5Cleft(%20%5Cfrac%7Bb%7D%7Ba%7D%20-%20%E2%88%86%20%5Cright)%20-%20%5Csqrt%7B%5Cleft(%20%5Cfrac%7Bb%7D%7Ba%7D%20-%20%E2%88%86%20%5Cright)%5E2-8%5Cleft(%20%E2%88%86_3%20-%20%20%5Cfrac%7Bb%E2%88%86_3-2d%7D%7Ba%E2%88%86%7D%20%5Cright)%7D%20%7D%7B4%7D%20

x_3%20%3D%20%5Cfrac%7B-%5Cleft(%20%5Cfrac%7Bb%7D%7Ba%7D%20%2B%20%E2%88%86%20%5Cright)%20%2B%20%5Csqrt%7B%5Cleft(%20%5Cfrac%7Bb%7D%7Ba%7D%20%2B%E2%88%86%20%5Cright)%5E2-8%5Cleft(%20%E2%88%86_3%20%2B%20%5Cfrac%7Bb%E2%88%86_3-2d%7D%7Ba%E2%88%86%7D%20%5Cright)%7D%20%7D%7B4%7D%20

x_4%20%3D%20%5Cfrac%7B-%5Cleft(%20%5Cfrac%7Bb%7D%7Ba%7D%20-%20%E2%88%86%20%5Cright)%20%2B%20%5Csqrt%7B%5Cleft(%20%5Cfrac%7Bb%7D%7Ba%7D%20-%20%E2%88%86%20%5Cright)%5E2-8%5Cleft(%20%E2%88%86_3%20-%20%5Cfrac%7Bb%E2%88%86_3-2d%7D%7Ba%E2%88%86%7D%20%5Cright)%7D%20%7D%7B4%7D%20

  其中:

  %E2%88%86_1%3Dc%5E2-3bd-12ae

  %E2%88%86_2%3D2c%5E3-9bcd%2B27ad%5E2%2B27b%5E2e-72ace

  %E2%88%86_3%3D%5Cfrac%7B%5Csqrt%5B3%5D%7B4%E2%88%86_2%2B4%5Csqrt%7B%E2%88%86_2%5E2-4%E2%88%86_1%5E3%7D%7D%2B%5Csqrt%5B3%5D%7B4%E2%88%86_2-4%5Csqrt%7B%E2%88%86_2%5E2-4%E2%88%86_1%5E3%7D%7D%7D%7B3a%7D%20

  %E2%88%86%3D%5Csqrt%7B%5Cfrac%7Bb%5E2-4ac%7D%7Ba%5E2%7D%20%2B%204%E2%88%86_3%7D%20

【原創(chuàng)】一元四次方程求根公式推導(dǎo)(可能有誤,敬請糾正)的評論 (共 條)

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