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Polynomial Equations[edit]
Linear Equations[edit]
![{\displaystyle ax\ +\ b\ =\ 0\ \qquad \qquad \qquad \qquad \quad ;\qquad x\ \ =\ -\ {\frac {b}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1251a17fc0c7646f62b04d8e50377f0410c1249c)
Quadratic Equations[edit]
![{\displaystyle ax^{2}\ +\ bx\ +\ a'\ =\ 0\ \qquad \qquad \quad \ ;\qquad x\ \ =\ -\ {\frac {1}{2a}}{\bigg (}b\ \pm \ {\sqrt {b^{2}\ -\ 4aa'}}{\bigg )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5839ead4ae290affac2997cc2d9d7393b3dc76bf)
Cubic Equations[edit]
![{\displaystyle 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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2ff164a3f39656662d0a28d800ad1781bf667d)
![{\displaystyle C\ \ =\ {\sqrt[{3\,}]{\frac {\Delta _{1}\ +\ {\sqrt {\Delta _{1}^{2}\ -\ 4\Delta _{0}^{3}}}}{2}}}\qquad ;\qquad {\color {white}.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c514a042d6c858b1ee78f7ec0969f7aa39f9f93c)
![{\displaystyle u_{_{\text{k}}}^{3}\ =\ 1\ \qquad ,\qquad \ k\in {\Big \{}1,\ 2,\ 3{\Big \}}\color {white}{\sqrt[{3}]{\frac {\sqrt {\Delta _{1}^{2}}}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c28690514a676f7a4c835003b72d72601f21847e)
![{\displaystyle \Delta _{0}=\ b^{2}-3ab^{'}\ \qquad \qquad \qquad \qquad ;\qquad \ \Delta _{1}=\ 2b^{3}-9a\,{\Big (}bb^{'}-3aa'{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7046da986d53ff5b51410ea5df5650d25510b82e)
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Observation:[edit]
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- The value of
has to be chosen so as to have C
0.
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Quartic Equations[edit]
![{\displaystyle ax^{4}\ +\ bx^{3}\ +\ cx^{2}\ +\ b^{'}x\ +\ a'\ =\ 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/046249ff990bcd7618c8270fd8a30960eca1c87d)
![{\displaystyle x\ =\ -\ {\frac {b}{4a}}\ \pm \ {\frac {\sqrt {u\ +\ v}}{2}}\ \ {\scriptstyle \bigoplus }\ \ {\frac {\sqrt {2u\ -\ v\ \mp \ w}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1248d7d6f6f8c2911f157e5383ce1f54922f880c)
![{\displaystyle 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}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be46b5565ccbf543497ba8722769159dd120e204)
![{\displaystyle Q\ \ =\ {\sqrt[{3\,}]{\frac {\Delta _{1}\ +\ {\sqrt {\Delta _{1}^{2}\ -\ 4\Delta _{0}^{3}}}}{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b41e70d53f3631c9b6e71db8d570e201c61847d)
![{\displaystyle \Delta _{0}\ =\ \ {\color {white}.}c^{2}\ -\ \ 3\,{\Big (}bb^{'}\ -\ 4aa'{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae1fe8354cf95e917f68fd4b0ff972d3dbddec5)
![{\displaystyle \Delta _{1}\ =\ 2c^{3}\ -\ 9c\,{\Big (}bb^{'}\ +\ 8aa'{\Big )}\ +\ 27\,{\Big (}a{\Big [}b^{'}{\Big ]}^{2}\ +\ a'b^{2}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af1f8c7473c4cdddc9fc4b8bc787372671d5560e)
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Observations:[edit]
——————————————————————————————————————————————————
- The value of
has to be chosen so as to have Q
0.
——————————————————————————————————————————————————
- The value of Q has to be chosen so as to have u + v
0.
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Fermat's Last Theorem[edit]
Pythagorean Triples[edit]
The Sum of Two Cubes[edit]
Generalization[edit]