『整合』一次到四次方程的通解

2022-10-03 21:05 作者:げいしも_芸 0人读过 | 我要投稿

注：所有方程系数均为实数

$ax%2Bb%3D0(a%5Cneq%200)$

$x%3D-%5Cfrac%20ba$

$ax%5E2%2Bbx%2Bc%3D0(a%5Cneq%200)$

$x_1%3D-%5Cfrac%7Bb%7D%7B2a%7D%2B%5Cfrac%7B%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D$

$x_2%3D-%5Cfrac%7Bb%7D%7B2a%7D-%5Cfrac%7B%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D$

$ax%5E3%2Bbx%5E2%2Bcx%2Bd%3D0(a%5Cneq%200)$

$%5Comega_1%3D%5Cfrac%7B-1%2B%5Csqrt%203%20i%7D%7B2%7D%2C%5Comega_2%3D%5Cfrac%7B-1-%5Csqrt%203%20i%7D%7B2%7D$

$u%3D%5Csqrt%5B3%5D%7B%5Cfrac%7Bbc%7D%7B6a%5E2%7D-%5Cfrac%7Bb%5E3%7D%7B27a%5E3%7D-%5Cfrac%7Bd%7D%7B2a%7D%2B%5Csqrt%7B(%5Cfrac%7Bbc%7D%7B6a%5E2%7D-%5Cfrac%7Bb%5E3%7D%7B27a%5E3%7D-%5Cfrac%7Bd%7D%7B2a%7D)%5E2%2B(%5Cfrac%20c%7B3a%7D-%5Cfrac%7Bb%5E2%7D%7B9a%5E2%7D)%5E3%7D%7D$

$v%3D%5Csqrt%5B3%5D%7B%5Cfrac%7Bbc%7D%7B6a%5E2%7D-%5Cfrac%7Bb%5E3%7D%7B27a%5E3%7D-%5Cfrac%7Bd%7D%7B2a%7D-%5Csqrt%7B(%5Cfrac%7Bbc%7D%7B6a%5E2%7D-%5Cfrac%7Bb%5E3%7D%7B27a%5E3%7D-%5Cfrac%7Bd%7D%7B2a%7D)%5E2%2B(%5Cfrac%20c%7B3a%7D-%5Cfrac%7Bb%5E2%7D%7B9a%5E2%7D)%5E3%7D%7D$

$x_1%3D-%5Cfrac%20b%7B3a%7D%2Bu%2Bv$

$x_2%3D-%5Cfrac%20b%7B3a%7D%2B%5Comega_1u%2B%5Comega_2v$

$x_3%3D-%5Cfrac%20b%7B3a%7D%2B%5Comega_2u%2B%5Comega_1v$

$ax%5E4%2Bbx%5E3%2Bcx%5E2%2Bdx%2Be%3D0(a%5Cneq%200%EF%BC%8Ce%E4%B8%8D%E4%B8%BA%E8%87%AA%E7%84%B6%E5%B8%B8%E6%95%B0)$

$p%3D%5Cfrac%7Bb%5E2%7D%7B8a%5E2%7D%2B%5Cfrac%20ca%2Cq%3D%5Cfrac%7Bb%5E2%7D%7B8a%5E2%7D-%5Cfrac%7Bbc%7D%7B2a%5E2%7D%2B%5Cfrac%20da%2Cr%3D-%5Cfrac%7B3b%5E4%7D%7B256a%5E4%7D%2B%5Cfrac%7Bb%5E2c%7D%7B16a%5E3%7D-%5Cfrac%7Bbd%7D%7B4a%5E2%7D%2B%5Cfrac%20ea$

$j%3D%5Csqrt%5B3%5D%7B-%5Cfrac%7Bpr%7D6%2B%5Cfrac%7Bp%5E3%7D%7B216%7D%2B%5Cfrac%7Bq%5E2%7D%7B16%7D%2B%5Csqrt%7B(-%5Cfrac%7Bpr%7D%7B6%7D%2B%5Cfrac%7Bp%5E3%7D%7B216%7D-%5Cfrac%7Bq%5E2%7D%7B16%7D)%5E2%2B(-%5Cfrac%20r3-%5Cfrac%7Bp%5E2%7D%7B36%7D)%5E3%7D%7D$

$k%3D%5Csqrt%5B3%5D%7B-%5Cfrac%7Bpr%7D6%2B%5Cfrac%7Bp%5E3%7D%7B216%7D%2B%5Cfrac%7Bq%5E2%7D%7B16%7D-%5Csqrt%7B(-%5Cfrac%7Bpr%7D%7B6%7D%2B%5Cfrac%7Bp%5E3%7D%7B216%7D-%5Cfrac%7Bq%5E2%7D%7B16%7D)%5E2%2B(-%5Cfrac%20r3-%5Cfrac%7Bp%5E2%7D%7B36%7D)%5E3%7D%7D$

$y%3D-%5Cfrac%20p3%2Bj%2Bk$

$x_1%3D-%5Cfrac%20b%7B4a%7D%2B%5Cfrac%2012%5Csqrt%7B-p%2B2y%7D%2B%5Cfrac%2012%5Csqrt%7B-p-2y%2B%5Cfrac%7B2q%7D%7Bp-2y%7D%5Csqrt%7B-p%2B2y%7D%7D$

$x_2%3D-%5Cfrac%20b%7B4a%7D%2B%5Cfrac%2012%5Csqrt%7B-p%2B2y%7D-%5Cfrac%2012%5Csqrt%7B-p-2y%2B%5Cfrac%7B2q%7D%7Bp-2y%7D%5Csqrt%7B-p%2B2y%7D%7D$

$x_3%3D-%5Cfrac%20b%7B4a%7D-%5Cfrac%2012%5Csqrt%7B-p%2B2y%7D%2B%5Cfrac%2012%5Csqrt%7B-p-2y-%5Cfrac%7B2q%7D%7Bp-2y%7D%5Csqrt%7B-p%2B2y%7D%7D$

$x_4%3D-%5Cfrac%20b%7B4a%7D-%5Cfrac%2012%5Csqrt%7B-p%2B2y%7D-%5Cfrac%2012%5Csqrt%7B-p-2y-%5Cfrac%7B2q%7D%7Bp-2y%7D%5Csqrt%7B-p%2B2y%7D%7D$

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