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- x³+8y=0
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-x³+8y=0 la ecuación
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Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$- x^{3} + 8 y = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 8 y = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - 8 y$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = - 8 y$$
$$- x^{3} + 8 y = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 8 y = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - 8 y$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = - 8 y$$
Gráfica
Suma y producto de raíces
[src]
suma
_________________ _________________ / _________________ _________________ \ _________________ _________________ / _________________ _________________ \ _________________ _________________
6 / 2 2 /atan2(im(y), re(y))\ 6 / 2 2 /atan2(im(y), re(y))\ | 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\ | 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\
2*\/ im (y) + re (y) *cos|-------------------| + 2*I*\/ im (y) + re (y) *sin|-------------------| + I*|- \/ im (y) + re (y) *sin|-------------------| - \/ 3 *\/ im (y) + re (y) *cos|-------------------|| - \/ im (y) + re (y) *cos|-------------------| + \/ 3 *\/ im (y) + re (y) *sin|-------------------| + I*|- \/ im (y) + re (y) *sin|-------------------| + \/ 3 *\/ im (y) + re (y) *cos|-------------------|| - \/ im (y) + re (y) *cos|-------------------| - \/ 3 *\/ im (y) + re (y) *sin|-------------------|
\ 3 / \ 3 / \ \ 3 / \ 3 // \ 3 / \ 3 / \ \ 3 / \ 3 // \ 3 / \ 3 /
$$\left(\left(2 i \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + 2 \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + \left(i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right)\right) + \left(i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right)$$
=
/ _________________ _________________ \ / _________________ _________________ \ _________________ | 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| | 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| 6 / 2 2 /atan2(im(y), re(y))\ I*|- \/ im (y) + re (y) *sin|-------------------| + \/ 3 *\/ im (y) + re (y) *cos|-------------------|| + I*|- \/ im (y) + re (y) *sin|-------------------| - \/ 3 *\/ im (y) + re (y) *cos|-------------------|| + 2*I*\/ im (y) + re (y) *sin|-------------------| \ \ 3 / \ 3 // \ \ 3 / \ 3 // \ 3 /
$$i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + 2 i \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}$$
producto
/ _________________ _________________ \ / / _________________ _________________ \ _________________ _________________ \ / / _________________ _________________ \ _________________ _________________ \ | 6 / 2 2 /atan2(im(y), re(y))\ 6 / 2 2 /atan2(im(y), re(y))\| | | 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| | | 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| |2*\/ im (y) + re (y) *cos|-------------------| + 2*I*\/ im (y) + re (y) *sin|-------------------||*|I*|- \/ im (y) + re (y) *sin|-------------------| - \/ 3 *\/ im (y) + re (y) *cos|-------------------|| - \/ im (y) + re (y) *cos|-------------------| + \/ 3 *\/ im (y) + re (y) *sin|-------------------||*|I*|- \/ im (y) + re (y) *sin|-------------------| + \/ 3 *\/ im (y) + re (y) *cos|-------------------|| - \/ im (y) + re (y) *cos|-------------------| - \/ 3 *\/ im (y) + re (y) *sin|-------------------|| \ \ 3 / \ 3 // \ \ \ 3 / \ 3 // \ 3 / \ 3 // \ \ \ 3 / \ 3 // \ 3 / \ 3 //
$$\left(2 i \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + 2 \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) \left(i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) \left(i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right)$$
=
/ / im(y) /atan2(im(y), re(y))\\ \
| 3*I*|-------------------- + sin|-------------------|| |
| /atan2(im(y), re(y))\ | _________________ \ 3 /| |
_________________ | 3*cos|-------------------| | / 2 2 | |
/ 2 2 | 3/atan2(im(y), re(y))\ \ 3 / 3/atan2(im(y), re(y))\ \\/ im (y) + re (y) / 3*re(y) |
8*\/ im (y) + re (y) *|cos |-------------------| - -------------------------- - I*sin |-------------------| + ----------------------------------------------------- + ----------------------|
| \ 3 / 4 \ 3 / 4 _________________|
| / 2 2 |
\ 4*\/ im (y) + re (y) /
$$8 \sqrt{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \left(\frac{3 i \left(\sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \frac{\operatorname{im}{\left(y\right)}}{\sqrt{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}\right)}{4} - i \sin^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \cos^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \frac{3 \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}}{4} + \frac{3 \operatorname{re}{\left(y\right)}}{4 \sqrt{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}\right)$$
8*sqrt(im(y)^2 + re(y)^2)*(cos(atan2(im(y), re(y))/3)^3 - 3*cos(atan2(im(y), re(y))/3)/4 - i*sin(atan2(im(y), re(y))/3)^3 + 3*i*(im(y)/sqrt(im(y)^2 + re(y)^2) + sin(atan2(im(y), re(y))/3))/4 + 3*re(y)/(4*sqrt(im(y)^2 + re(y)^2)))
Respuesta rápida
[src]
_________________ _________________
6 / 2 2 /atan2(im(y), re(y))\ 6 / 2 2 /atan2(im(y), re(y))\
x1 = 2*\/ im (y) + re (y) *cos|-------------------| + 2*I*\/ im (y) + re (y) *sin|-------------------|
\ 3 / \ 3 /
$$x_{1} = 2 i \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + 2 \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}$$
/ _________________ _________________ \ _________________ _________________
| 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\
x2 = I*|- \/ im (y) + re (y) *sin|-------------------| - \/ 3 *\/ im (y) + re (y) *cos|-------------------|| - \/ im (y) + re (y) *cos|-------------------| + \/ 3 *\/ im (y) + re (y) *sin|-------------------|
\ \ 3 / \ 3 // \ 3 / \ 3 /
$$x_{2} = i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}$$
/ _________________ _________________ \ _________________ _________________
| 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\| 6 / 2 2 /atan2(im(y), re(y))\ ___ 6 / 2 2 /atan2(im(y), re(y))\
x3 = I*|- \/ im (y) + re (y) *sin|-------------------| + \/ 3 *\/ im (y) + re (y) *cos|-------------------|| - \/ im (y) + re (y) *cos|-------------------| - \/ 3 *\/ im (y) + re (y) *sin|-------------------|
\ \ 3 / \ 3 // \ 3 / \ 3 /
$$x_{3} = i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}$$
x3 = i*(-(re(y)^2 + im(y)^2)^(1/6)*sin(atan2(im(y, re(y))/3) + sqrt(3)*(re(y)^2 + im(y)^2)^(1/6)*cos(atan2(im(y), re(y))/3)) - sqrt(3)*(re(y)^2 + im(y)^2)^(1/6)*sin(atan2(im(y), re(y))/3) - (re(y)^2 + im(y)^2)^(1/6)*cos(atan2(im(y), re(y))/3))