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三次方程式、 a x 3 + b x 2 + c x + d {\displaystyle ax^{3}+bx^{2}+cx+d\,\!} のp,qを展開した解です。
x = 1 6 a ( − 108 a 2 d + 36 a b c − 8 b 3 + 4 b 3 ( 4 b 3 − 1 ) + 36 a b 2 c ( 1 − b 2 ) + 27 a 2 b ( 3 b c 2 + 4 b 2 d − 4 c 2 ) + 18 a 3 c ( 2 c 2 − 27 b d ) + 729 a 4 d 2 3 + − 108 a 2 d + 36 a b c − 8 b 3 − 4 b 3 ( 4 b 3 − 1 ) + 36 a b 2 c ( 1 − b 2 ) + 27 a 2 b ( 3 b c 2 + 4 b 2 d − 4 c 2 ) + 18 a 3 c ( 2 c 2 − 27 b d ) + 729 a 4 d 2 3 − 2 b ) {\displaystyle x={\frac {1}{6a}}\left({\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}+4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}+{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}-4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}-2b\right)}
x = 1 6 a ( ω − 108 a 2 d + 36 a b c − 8 b 3 + 4 b 3 ( 4 b 3 − 1 ) + 36 a b 2 c ( 1 − b 2 ) + 27 a 2 b ( 3 b c 2 + 4 b 2 d − 4 c 2 ) + 18 a 3 c ( 2 c 2 − 27 b d ) + 729 a 4 d 2 3 + ω 2 − 108 a 2 d + 36 a b c − 8 b 3 − 4 b 3 ( 4 b 3 − 1 ) + 36 a b 2 c ( 1 − b 2 ) + 27 a 2 b ( 3 b c 2 + 4 b 2 d − 4 c 2 ) + 18 a 3 c ( 2 c 2 − 27 b d ) + 729 a 4 d 2 3 − 2 b ) {\displaystyle x={\frac {1}{6a}}\left(\omega {\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}+4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}+\omega ^{2}{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}-4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}-2b\right)}
x = 1 6 a ( ω 2 − 108 a 2 d + 36 a b c − 8 b 3 + 4 b 3 ( 4 b 3 − 1 ) + 36 a b 2 c ( 1 − b 2 ) + 27 a 2 b ( 3 b c 2 + 4 b 2 d − 4 c 2 ) + 18 a 3 c ( 2 c 2 − 27 b d ) + 729 a 4 d 2 3 + ω − 108 a 2 d + 36 a b c − 8 b 3 − 4 b 3 ( 4 b 3 − 1 ) + 36 a b 2 c ( 1 − b 2 ) + 27 a 2 b ( 3 b c 2 + 4 b 2 d − 4 c 2 ) + 18 a 3 c ( 2 c 2 − 27 b d ) + 729 a 4 d 2 3 − 2 b ) {\displaystyle x={\frac {1}{6a}}\left(\omega ^{2}{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}+4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}+\omega {\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}-4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}-2b\right)}
注1：誤植がある可能性はあります。 注2：そのまま印刷するとはみ出ると思うので、印刷用にはこれを使って下さい：
x = 1 6 a ( − 108 a 2 d + 36 a b c − 8 b 3 + 4 n 3 + − 108 a 2 d + 36 a b c − 8 b 3 − 4 n 3 − 2 b ) {\displaystyle x={\frac {1}{6a}}\left(\,\,{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}+4{\sqrt {\rm {n}}}}}\,\,+\;\;\;{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}-4{\sqrt {\rm {n}}}}}-2b\right)}
x = 1 6 a ( ω − 108 a 2 d + 36 a b c − 8 b 3 + 4 n 3 + ω 2 − 108 a 2 d + 36 a b c − 8 b 3 − 4 n 3 − 2 b ) {\displaystyle x={\frac {1}{6a}}\left(\omega {\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}+4{\sqrt {\rm {n}}}}}+\omega ^{2}{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}-4{\sqrt {\rm {n}}}}}-2b\right)}
x = 1 6 a ( ω 2 − 108 a 2 d + 36 a b c − 8 b 3 + 4 n 3 + ω − 108 a 2 d + 36 a b c − 8 b 3 − 4 n 3 − 2 b ) {\displaystyle x={\frac {1}{6a}}\left(\omega ^{2}{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}+4{\sqrt {\rm {n}}}}}+\omega {\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}-4{\sqrt {\rm {n}}}}}-2b\right)} b 3 ( 4 b 3 − 1 ) + 36 a b 2 c ( 1 − b 2 ) + 27 a 2 b ( 3 b c 2 + 4 b 2 d − 4 c 2 ) + 18 a 3 c ( 2 c 2 − 27 b d ) + 729 a 4 d 2 = n {\displaystyle \,\!b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}={\rm {n}}}
のp,qを展開した解です。
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![{\displaystyle x={\frac {1}{6a}}\left(\omega {\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}+4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}+\omega ^{2}{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}-4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}-2b\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89d5a205b5a06ff3c63d35af9c7e63dce0245632)
![{\displaystyle x={\frac {1}{6a}}\left(\omega ^{2}{\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}+4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}+\omega {\sqrt[{3}]{-108a^{2}d+36abc-8b^{3}-4{\sqrt {b^{3}(4b^{3}-1)+36ab^{2}c(1-b^{2})+27a^{2}b(3bc^{2}+4b^{2}d-4c^{2})+18a^{3}c(2c^{2}-27bd)+729a^{4}d^{2}}}}}-2b\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4afd5dcfd3a5dc8f1edd6f860c2cd869ab53ba25)
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