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Solver for Octic Equations Solver for Decic Equations |

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Formula for Octic Equations

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Complete Formula

Please read it from the PDF (Traditional Chinese): Formula08 (215.4KB, click the link to download or open in the Explorer)

If you cannot understand Chinese, you can download the English version at here, but it is written in plain text, you should open it after installing and choosing the font "TH-Formulae-Ext". (Since the formulae were updated, the ETXT cannot show the new formulae correctly, so the ETXT files on this site is still the old version, if you want to see the new version, please download the Chinese version, sorry --2020.02.08)

TH-Theorem

(1)Discriminant Rules

Only when H=K=M=0 or both P=Q=R=0 and S>0, the equation can be solved by TH-Formula(Octic). Some equations which are not satisfied with the condition can meet the condition after some transformations.

If an equation can be solved by TH-Formulae, repeated roots exist when Δ=0.

(2)Others

<1>Imaginary numbers will not appear in any radical signs in the formula.

Examples

Click the following buttons to choose the examples in different classifications. No example for Formula1, while two examples for every other formula.

For convenient calculating, the octic coefficients are all 1.

Click this link to download all the examples (including Decic Equations); click this link download the answer.

Directory

1.Complete Formula 2.TH-Theorem 3.Examples |

Complete Formula

Please read it from the PDF (Traditional Chinese): Formula08 (215.4KB, click the link to download or open in the Explorer)

If you cannot understand Chinese, you can download the English version at here, but it is written in plain text, you should open it after installing and choosing the font "TH-Formulae-Ext". (Since the formulae were updated, the ETXT cannot show the new formulae correctly, so the ETXT files on this site is still the old version, if you want to see the new version, please download the Chinese version, sorry --2020.02.08)

TH-Theorem

(1)Discriminant Rules

Only when H=K=M=0 or both P=Q=R=0 and S>0, the equation can be solved by TH-Formula(Octic). Some equations which are not satisfied with the condition can meet the condition after some transformations.

If an equation can be solved by TH-Formulae, repeated roots exist when Δ=0.

(2)Others

<1>Imaginary numbers will not appear in any radical signs in the formula.

Examples

Click the following buttons to choose the examples in different classifications. No example for Formula1, while two examples for every other formula.

For convenient calculating, the octic coefficients are all 1.

Click this link to download all the examples (including Decic Equations); click this link download the answer.