一元二、三、四次方程求根公式、判别式、韦达定理、已知方程根求作方程

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一元二、三、四次方程求根公式、判别式、韦达定理、已知方程根求作方程

2024-07-09 22:57| 来源: 网络整理| 查看: 265

下面是我整理的一元二、三、四次方程求根公式、判别式、韦达定理、已知方程根求作方程(韦达定理逆定理)内容。

一元二次方程

$ax%5E2%2Bbx%2Bc%3D0(a%E2%89%A00%2Ca%2Cb%2Cc%E2%88%88%5Cmathbb%20R)$

求根公式:

$%5Cbegin%7Bmatrix%7Dx_1%3D%5Cdfrac%7B-b%2B%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%2C%5C%5Cx_2%3D%5Cdfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D.%5Cend%7Bmatrix%7D$

判别式:

$%CE%94%3Db%5E2-4ac.$

当 $%CE%94%3E0$ 时,方程有两个不等实根.

当 $%CE%94%3D0$ 时,方程有一个二重实根

$x_1%3Dx_2%3D-%5Cfrac%20b%7B2a%7D.$

当 $%CE%94%3C0$ 时,方程有一对共轭虚根.

韦达定理:

$%5Cbegin%7Bmatrix%7Dx_1%2Bx_2%3D-%5Cdfrac%20ba%2C%5C%5Cx_1x_2%3D%5Cdfrac%20ca.%5Cend%7Bmatrix%7D$

以 $x_1%2Cx_2$ 为两根的一元二次方程:

$x%5E2-(x_1%2Bx_2)x%2Bx_1x_2%3D0.$

一元三次方程

$ax%5E3%2Bbx%5E2%2Bcx%2Bd%3D0(a%E2%89%A00%2Ca%2Cb%2Cc%2Cd%E2%88%88%5Cmathbb%20R)$

求根公式:

$%5Cbegin%7Bmatrix%7Dx_1%3D%5Cdfrac%7B-b%2B%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d%2B3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%2B%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d-3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%7D%7B3a%7D%2C%5C%5Cx_2%3D%5Cdfrac%7B-b%2B%5Cdfrac%7B-1%2B%5Csqrt3%5Ctext%20i%7D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d%2B3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%2B%5Cdfrac%7B-1-%5Csqrt3%5Ctext%20i%7D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d-3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%7D%7B3a%7D%2C%5C%5Cx_3%3D%5Cdfrac%7B-b%2B%5Cdfrac%7B-1-%5Csqrt3%5Ctext%20i%7D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d%2B3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%2B%5Cdfrac%7B-1%2B%5Csqrt3%5Ctext%20i%7D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d-3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%7D%7B3a%7D.%5Cend%7Bmatrix%7D$

判别式:

$%CE%94%3D-b%5E2c%5E2%2B4ac%5E3%2B4b%5E3d-18abcd%2B27a%5E2d%5E2.%0A%0A$

当 $%CE%94%3E0$ 时,方程有一个实根和一对共轭虚根.

当 $%CE%94%3D0%2Cb%5E2-3ac%E2%89%A00$ 时,方程有三个实根,其中有一个二重根

$%5Cbegin%7Bmatrix%7Dx_1%3D%5Cdfrac%7B-b-2%5Coperatorname%7Bsgn%7D(2b%5E3-9abc%2B27a%5E2d)%5Csqrt%7Bb%5E2-3ac%7D%7D%7B3a%7D%2C%5C%5Cx_2%3Dx_3%3D%5Cdfrac%7B-b%2B%5Coperatorname%7Bsgn%7D(2b%5E3-9abc%2B27a%5E2d)%5Csqrt%7Bb%5E2-3ac%7D%7D%7B3a%7D.%5Cend%7Bmatrix%7D$

当 $%CE%94%3Db%5E2-3ac%3D0$ 时,方程有一个三重实根

$x_1%3Dx_2%3Dx_3%3D-%5Cfrac%20b%7B3a%7D.$

当 $%CE%94%3C0$ 时,方程有三个不等实根.

韦达定理:

$%5Cbegin%7Bmatrix%7Dx_1%2Bx_2%2Bx_3%3D-%5Cdfrac%20ba%2C%5C%5Cx_1x_2%2Bx_1x_3%2Bx_2x_3%3D%5Cdfrac%20ca%2C%5C%5Cx_1x_2x_3%3D-%5Cdfrac%20da.%5Cend%7Bmatrix%7D$

以 $x_1%2Cx_2%2Cx_3$ 为三根的一元三次方程:

$x%5E3-(x_1%2Bx_2%2Bx_3)x%5E2%2B(x_1x_2%2Bx_1x_3%2Bx_2x_3)x-x_1x_2x_3%3D0.$

一元四次方程

$ax%5E4%2Bbx%5E3%2Bcx%5E2%2Bdx%2Be%3D0(a%E2%89%A00%2Ca%2Cb%2Cc%2Cd%2Ce%E2%88%88%5Cmathbb%20R)$

求根公式:

$k%5E3-ck%5E2%2B(bd-4ae)k-ad%5E2-b%5E2e%2B4ace%3D0%2C$

当 $b%5E3-4abc%2B8a%5E2d%3E0$ 时,

$%5Cbegin%7Bmatrix%7Dx_1%3D%5Cdfrac%7B-b%2B%5Csqrt%7Bb%5E2-4ac%2B4ak%7D%2B%5Csqrt%7B2b%5E2-4ac-4ak-2%5Csqrt%7Bb%5E4-4ab%5E2c-4ab%5E2k%2B16a%5E2bd%2B16a%5E2k%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_2%3D%5Cdfrac%7B-b%2B%5Csqrt%7Bb%5E2-4ac%2B4ak%7D-%5Csqrt%7B2b%5E2-4ac-4ak-2%5Csqrt%7Bb%5E4-4ab%5E2c-4ab%5E2k%2B16a%5E2bd%2B16a%5E2k%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_3%3D%5Cdfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%2B4ak%7D%2B%5Csqrt%7B2b%5E2-4ac-4ak%2B2%5Csqrt%7Bb%5E4-4ab%5E2c-4ab%5E2k%2B16a%5E2bd%2B16a%5E2k%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_4%3D%5Cdfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%2B4ak%7D-%5Csqrt%7B2b%5E2-4ac-4ak%2B2%5Csqrt%7Bb%5E4-4ab%5E2c-4ab%5E2k%2B16a%5E2bd%2B16a%5E2k%5E2-64a%5E3e%7D%7D%7D%7B4a%7D.%5Cend%7Bmatrix%7D$

当 $b%5E3-4abc%2B8a%5E2d%3D0$ 时,

$%5Cbegin%7Bmatrix%7Dx_1%3D%5Cdfrac%7B-b%2B%5Csqrt%7B3b%5E2-8ac%2B2%5Csqrt%7B3b%5E4-16ab%5E2c%2B16a%5E2bd%2B16a%5E2c%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_2%3D%5Cdfrac%7B-b-%5Csqrt%7B3b%5E2-8ac%2B2%5Csqrt%7B3b%5E4-16ab%5E2c%2B16a%5E2bd%2B16a%5E2c%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_3%3D%5Cdfrac%7B-b%2B%5Csqrt%7B3b%5E2-8ac-2%5Csqrt%7B3b%5E4-16ab%5E2c%2B16a%5E2bd%2B16a%5E2c%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_4%3D%5Cdfrac%7B-b-%5Csqrt%7B3b%5E2-8ac-2%5Csqrt%7B3b%5E4-16ab%5E2c%2B16a%5E2bd%2B16a%5E2c%5E2-64a%5E3e%7D%7D%7D%7B4a%7D.%5Cend%7Bmatrix%7D$

当 $b%5E3-4abc%2B8a%5E2d%3C0$ 时,

$%5Cbegin%7Bmatrix%7Dx_1%3D%5Cdfrac%7B-b%2B%5Csqrt%7Bb%5E2-4ac%2B4ak%7D%2B%5Csqrt%7B2b%5E2-4ac-4ak%2B2%5Csqrt%7Bb%5E4-4ab%5E2c-4ab%5E2k%2B16a%5E2bd%2B16a%5E2k%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_2%3D%5Cdfrac%7B-b%2B%5Csqrt%7Bb%5E2-4ac%2B4ak%7D-%5Csqrt%7B2b%5E2-4ac-4ak%2B2%5Csqrt%7Bb%5E4-4ab%5E2c-4ab%5E2k%2B16a%5E2bd%2B16a%5E2k%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_3%3D%5Cdfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%2B4ak%7D%2B%5Csqrt%7B2b%5E2-4ac-4ak-2%5Csqrt%7Bb%5E4-4ab%5E2c-4ab%5E2k%2B16a%5E2bd%2B16a%5E2k%5E2-64a%5E3e%7D%7D%7D%7B4a%7D%2C%5C%5Cx_4%3D%5Cdfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%2B4ak%7D-%5Csqrt%7B2b%5E2-4ac-4ak-2%5Csqrt%7Bb%5E4-4ab%5E2c-4ab%5E2k%2B16a%5E2bd%2B16a%5E2k%5E2-64a%5E3e%7D%7D%7D%7B4a%7D.%5Cend%7Bmatrix%7D$

判别式:

$%CE%94%3D-b%5E2c%5E2d%5E2%2B4ac%5E3d%5E2%2B4b%5E3d%5E3%2B4b%5E2c%5E3e%2B6ab%5E2d%5E2e-16ac%5E4e-18abcd%5E3-18b%5E3cde%2B27a%5E2d%5E4%2B27b%5E4e%5E2%2B80abc%5E2de%2B128a%5E2c%5E2e%5E2-144a%5E2cd%5E2e-144ab%5E2ce%5E2%2B192a%5E2bde%5E2-256a%5E3e%5E3.$

当 $%CE%94%3E0$ 时,方程有两个不等实根和一对共轭虚根.

当 $%CE%94%3D0%2Cc%5E2-3bd%2B12ae%E2%89%A00$ 时,方程有一个二重实根;若 $b%5E2c%5E2-3b%5E3d-4ac%5E3-6ab%5E2e%2B14abcd%2B16a%5E2ce-18a%5E2d%5E2%3E0$ ,则其余两根为不等实根;若 $b%5E2c%5E2-3b%5E3d-4ac%5E3-6ab%5E2e%2B14abcd%2B16a%5E2ce-18a%5E2d%5E2%3C0$ ,则其余两根为共轭虚根

$%5Cbegin%7Bmatrix%7Dx_%7B1%2C2%7D%3D%5Cdfrac%7B-b-%5Coperatorname%7Bsgn%7D(b%5E3-4abc%2B8a%5E2d)%5Csqrt%7B%5Cdfrac%7B3b%5E2-8ac%2B8a%5Coperatorname%7Bsgn%7D(2c%5E3-9bcd%2B27ad%5E2%2B27b%5E2e-72ace)%5Csqrt%7Bc%5E2-3bd%2B12ae%7D%7D3%7D%5Cpm2%5Csqrt%7B%5Cdfrac%7B3b%5E2-8ac-4a%5Coperatorname%7Bsgn%7D(2c%5E3-9bcd%2B27ad%5E2%2B27b%5E2e-72ace)%5Csqrt%7Bc%5E2-3bd%2B12ae%7D%7D3%7D%7D%7B4a%7D%2C%5C%5Cx_3%3Dx_4%3D%5Cdfrac%7B-b%2B%5Coperatorname%7Bsgn%7D(b%5E3-4abc%2B8a%5E2d)%5Csqrt%7B%5Cdfrac%7B3b%5E2-8ac%2B8a%5Coperatorname%7Bsgn%7D(2c%5E3-9bcd%2B27ad%5E2%2B27b%5E2e-72ace)%5Csqrt%7Bc%5E2-3bd%2B12ae%7D%7D3%7D%7D%7B4a%7D.%5Cend%7Bmatrix%7D$

当 $%CE%94%3Dc%5E2-3bd%2B12ae%3D0%2C3b%5E2-8ac%E2%89%A00$ 时,方程有四个实根,其中有一个三重根

$%5Cbegin%7Bmatrix%7Dx_1%3D%5Cdfrac%7B-b-3%5Csqrt%5B3%5D%7Bb%5E3-4abc%2B8a%5E2d%7D%7D%7B4a%7D%2C%5C%5Cx_2%3Dx_3%3Dx_4%3D%5Cdfrac%7B-b%2B%5Csqrt%5B3%5D%7Bb%5E3-4abc%2B8a%5E2d%7D%7D%7B4a%7D.%5Cend%7Bmatrix%7D$

当 $%CE%94%3D3b%5E4-16ab%5E2c%2B16a%5E2bd%2B16a%5E2c%5E2-64a%5E3e%3D0%2C3b%5E2-8ac%E2%89%A00$ 时,方程有两个二重根;若 $3b%5E2-8ac%3E0$ ,根为实数;若 $3b%5E2-8ac%3C0$ ,根为虚数

$x_%7B1%2C2%7D%3Dx_%7B3%2C4%7D%3D%5Cfrac%7B-b%5Cpm%5Csqrt%7B3b%5E2-8ac%7D%7D%7B4a%7D.$

当 $%CE%94%3Dc%5E2-3bd%2B12ae%3D3b%5E2-8ac%3D0$ 时,方程有一个四重实根

$x_1%3Dx_2%3Dx_3%3Dx_4%3D-%5Cfrac%20b%7B4a%7D.$

当 $%CE%94%3C0$ 时,若 $3b%5E2-8ac%3E0%2C3b%5E4-16ab%5E2c%2B16a%5E2bd%2B16a%5E2c%5E2-64a%5E3e%3E0$ ,则方程有四个不等实根;否则方程有两对不等共轭虚根.

韦达定理:

$%5Cbegin%7Bmatrix%7Dx_1%2Bx_2%2Bx_3%2Bx_4%3D-%5Cdfrac%20ba%2C%5C%5Cx_1x_2%2Bx_1x_3%2Bx_1x_4%2Bx_2x_3%2Bx_2x_4%2Bx_3x_4%3D%5Cdfrac%20ca%2C%5C%5Cx_1x_2x_3%2Bx_1x_2x_4%2Bx_1x_3x_4%2Bx_2x_3x_4%3D-%5Cdfrac%20da%2C%5C%5Cx_1x_2x_3x_4%3D%5Cdfrac%20ea.%5Cend%7Bmatrix%7D$

以 $x_1%2Cx_2%2Cx_3%2Cx_4$ 为四根的一元四次方程:

$x%5E4-(x_1%2Bx_2%2Bx_3%2Bx_4)x%5E3%2B(x_1x_2%2Bx_1x_3%2Bx_1x_4%2Bx_2x_3%2Bx_2x_4%2Bx_3x_4)x%5E2-(x_1x_2x_3%2Bx_1x_2x_4%2Bx_1x_3x_4%2Bx_2x_3x_4)x%2Bx_1x_2x_3x_4%3D0.$

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一元二、三、四次方程求根公式、判别式、韦达定理、已知方程根求作方程

一元二、三、四次方程求根公式、判别式、韦达定理、已知方程根求作方程

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