一元二、三、四次方程求根公式、判別式、韋達(dá)定理、已知方程根求作方程

2023-03-17 04:14 作者:超級安安喵 0人讀過 | 我要投稿

下面是我整理的一元二、三、四次方程求根公式、判別式、韋達(dá)定理、已知方程根求作方程(韋達(dá)定理逆定理)內(nèi)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:

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

判別式:

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

當(dāng) $%CE%94%3E0$ 時,方程有兩個不等實根.

當(dāng) $%CE%94%3D0$ 時,方程有一個二重實根

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

當(dāng) $%CE%94%3C0$ 時,方程有一對共軛虛根.

韋達(dá)定理:

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

以 $x_1%2Cx_2$ 為兩根的一元二次方程:

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

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:

$%5Cbegin%7Bmatrix%7Dx_1%3D%5Cdfrac%7B-b%2B%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d%2B3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%2B%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d-3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%7D%7B3a%7D%2C%5C%5Cx_2%3D%5Cdfrac%7B-b%2B%5Cdfrac%7B-1%2B%5Csqrt3i%7D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d%2B3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%2B%5Cdfrac%7B-1-%5Csqrt3i%7D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d-3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%7D%7B3a%7D%2C%5C%5Cx_3%3D%5Cdfrac%7B-b%2B%5Cdfrac%7B-1-%5Csqrt3i%7D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d%2B3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%2B%5Cdfrac%7B-1%2B%5Csqrt3i%7D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B-2b%5E3%2B9abc-27a%5E2d-3a%5Csqrt%7B-3b%5E2c%5E2%2B12ac%5E3%2B12b%5E3d-54abcd%2B81a%5E2d%5E2%7D%7D2%7D%7D%7B3a%7D.%5Cend%7Bmatrix%7D$

判別式:

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

當(dāng) $%CE%94%3E0$ 時,方程有一個實根和一對共軛虛根.

當(dāng) $%CE%94%3D0%2Cb%5E2-3ac%E2%89%A00$ 時,方程有一對二重實根

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

當(dāng) $%CE%94%3Db%5E2-3ac%3D0$ 時,方程有一個三重實根

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

當(dāng) $%CE%94%3C0$ 時,方程有三個不等實根.

韋達(dá)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 $x_1%2Cx_2%2Cx_3$ 為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:

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

當(dāng) $b%5E3-4abc%2B8a%5E2d%3E0$ 時,

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

當(dāng) $b%5E3-4abc%2B8a%5E2d%3D0$ 時,

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

當(dāng) $b%5E3-4abc%2B8a%5E2d%3C0$ 時,

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

判別式:

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

當(dāng) $%CE%94%3E0$ 時,方程有兩個不等實根和一對共軛虛根.

當(dāng) $%CE%94%3D0%2Cc%5E2-3bd%2B12ae%E2%89%A00$ 時,方程有一對二重實根;若 $b%5E2c%5E2-3b%5E3d-4ac%5E3-6ab%5E2e%2B14abcd%2B16a%5E2ce-18a%5E2d%5E2%3E0$ ,則其余兩根為不等實根;若 $b%5E2c%5E2-3b%5E3d-4ac%5E3-6ab%5E2e%2B14abcd%2B16a%5E2ce-18a%5E2d%5E2%3C0$ ,則其余兩根為共軛虛根

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

當(dāng) $%CE%94%3Dc%5E2-3bd%2B12ae%3D0%2C3b%5E2-8ac%E2%89%A00$ 時,方程有四個實根,其中有一個三重根

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

當(dāng) $%CE%94%3D3b%5E4%2B16a%5E2c%5E2-16ab%5E2c%2B16a%5E2bd-64a%5E3e%3D0%2C3b%5E2-8ac%E2%89%A00$ 時,方程有兩對二重根;若 $3b%5E2-8ac%3E0$ ,根為實數(shù);若 $3b%5E2-8ac%3C0$ ,根為虛數(shù)

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

當(dāng) $%CE%94%3Dc%5E2-3bd%2B12ae%3D3b%5E2-8ac%3D0$ 時,方程有一個四重實根

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

當(dāng) $%CE%94%3C0$ 時,若 $3b%5E2-8ac%3E0%2C3b%5E4%2B16a%5E2c%5E2-16ab%5E2c%2B16a%5E2bd-64a%5E3e%3E0$ ,則方程有四個不等實根;否則方程有兩對不等共軛虛根.

韋達(dá)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 $x_1%2Cx_2%2Cx_3%2Cx_4%E4%B8%BA$ 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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