User:GX, May 1971/Math/Algebra

Polynomial Equations

Linear Equations

$ax\ +\ b\ =\ 0\ \qquad \qquad \qquad \qquad \quad ;\qquad x\ \ =\ -\ {\frac {b}{a}}$

Quadratic Equations

$ax^{2}\ +\ bx\ +\ a'\ =\ 0\ \qquad \qquad \quad \ ;\qquad x\ \ =\ -\ {\frac {1}{2a}}{\bigg (}b\ \pm \ {\sqrt {b^{2}\ -\ 4aa'}}{\bigg )}$

Cubic Equations

$ax^{3}\ +\ bx^{2}\ +\ b^{'}x\ +\ a'\ =\ 0\qquad ;\qquad x_{_{\text{k}}}\ =\ -\ {\frac {1}{3a}}{\Bigg (}b\ +\ u_{_{\text{k}}}C\ +\ {\frac {\Delta _{0}}{u_{_{\text{k}}}C}}{\Bigg )}$

$C\ \ =\ {\sqrt[{3\,}]{\frac {\Delta _{1}\ +\ {\sqrt {\Delta _{1}^{2}\ -\ 4\Delta _{0}^{3}}}}{2}}}\qquad ;\qquad {\color {white}.}$$u_{_{\text{k}}}^{3}\ =\ 1\ \qquad ,\qquad \ k\in {\Big \{}1,\ 2,\ 3{\Big \}}\color {white}{\sqrt[{3}]{\frac {\sqrt {\Delta _{1}^{2}}}{2}}}$

$\Delta _{0}=\ b^{2}-3ab^{'}\ \qquad \qquad \qquad \qquad ;\qquad \ \Delta _{1}=\ 2b^{3}-9a\,{\Big (}bb^{'}-3aa'{\Big )}$

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Observation:

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The value of ${\color {white}.}{\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}\ {\color {white}.}$ has to be chosen so as to have C $\scriptstyle \neq$ 0.

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Quartic Equations

$ax^{4}\ +\ bx^{3}\ +\ cx^{2}\ +\ b^{'}x\ +\ a'\ =\ 0$

$x\ =\ -\ {\frac {b}{4a}}\ \pm \ {\frac {\sqrt {u\ +\ v}}{2}}\ \ {\scriptstyle \bigoplus }\ \ {\frac {\sqrt {2u\ -\ v\ \mp \ w}}{2}}$

$u\ =\ {\frac {3b^{2}-8ac}{12a^{2}}}\quad ;\quad v\ =\ {\frac {1}{3a}}{\Bigg (}Q\ +\ {\frac {\Delta _{0}}{Q}}{\Bigg )}\quad ;\quad w\ =\ {\frac {b^{3}\ -\ 4abc\ +\ 8a^{2}b^{'}}{4a^{3}\,{\sqrt {u\ +\ v}}}}$

$Q\ \ =\ {\sqrt[{3\,}]{\frac {\Delta _{1}\ +\ {\sqrt {\Delta _{1}^{2}\ -\ 4\Delta _{0}^{3}}}}{2}}}$

$\Delta _{0}\ =\ \ {\color {white}.}c^{2}\ -\ \ 3\,{\Big (}bb^{'}\ -\ 4aa'{\Big )}$

$\Delta _{1}\ =\ 2c^{3}\ -\ 9c\,{\Big (}bb^{'}\ +\ 8aa'{\Big )}\ +\ 27\,{\Big (}a{\Big [}b^{'}{\Big ]}^{2}\ +\ a'b^{2}{\Big )}$

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Observations:

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The value of ${\color {white}.}{\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}\ {\color {white}.}$ has to be chosen so as to have Q $\scriptstyle \neq$ 0.

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The value of Q has to be chosen so as to have u + v $\scriptstyle \neq$ 0.

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