User:GX, May 1971/Math/Algebra

= $Polynomial Equations$ =

$Linear Equations$

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

$Quadratic Equations$

 * $$a x^2\ +\ b x\ +\ a'\ =\ 0\ \qquad\qquad\quad\ ;\qquad x\ \ =\ -\ \frac1{2a}\bigg(b\ \pm\ \sqrt{b^2\ -\ 4aa'}\bigg)$$

$Cubic Equations$

 * $$a x^3\ +\ b x^2\ +\ b^' x\ +\ a'\ =\ 0 \qquad;\qquad x_{_\text{k}}\ =\ -\ \frac1{3a}\Bigg(b\ +\ u_{_\text{k}} C\ +\ \frac{\Delta_0}{u_{_\text{k}} C}\Bigg)$$


 * $$C\ \ =\ \sqrt[3\,]\frac{\Delta_1\ +\ \sqrt{\Delta_1^2\ -\ 4 \Delta_0^3}}{2} \qquad;\qquad {\color{white}.}$$$$u_{_\text{k}}^3\ =\ 1\ \qquad,\qquad\ k \in \Big\{1,\ 2,\ 3\Big\} \color{white} \sqrt[3]\frac\sqrt{\Delta_1^2}{2}$$


 * $$\Delta_0 =\ b^2 - 3ab^'\ \qquad\qquad\qquad\qquad;\qquad\ \Delta_1 =\ 2b^3 - 9a\,\Big(bb^' - 3aa'\Big)$$

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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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$Quartic Equations$

 * $$a x^4\ +\ b x^3\ +\ c x^2\ +\ b^' x\ +\ a'\ =\ 0$$


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


 * $$u\ =\ \frac{3b^2 - 8ac}{12a^2} \quad;\quad v\ =\ \frac1{3a}\Bigg(Q\ +\ \frac{\Delta_0}{Q}\Bigg) \quad;\quad w\ =\ \frac{b^3\ -\ 4abc\ +\ 8a^2b^'}{4a^3\,\sqrt{u\ +\ v}}$$


 * $$Q\ \ =\ \sqrt[3\,]\frac{\Delta_1\ +\ \sqrt{\Delta_1^2\ -\ 4 \Delta_0^3}}{2}$$


 * $$\Delta_0\ =\ \ {\color{white}.} c^2\ -\ \ 3\,\Big(bb^'\ -\ 4aa'\Big)$$


 * $$\Delta_1\ =\ 2c^3\ -\ 9c\,\Big(bb^'\ +\ 8aa'\Big)\ +\ 27\,\Big(a\Big[b^'\Big]^2\ +\ a'b^2\Big)$$

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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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= $Fermat's Last Theorem$ =