Sr Examen
x^3-y^2=8 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - y^{2} - 8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = - y^{2} - 8$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - y^{2} - 8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = - y^{2} - 8$$
Gráfica
Respuesta rápida
[src]
__________________________________________ __________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|
x1 = \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| + I*\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|
\ 3 / \ 3 /
$$x_{1} = i \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)} + \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}$$
/ __________________________________________ __________________________________________ \ __________________________________________ __________________________________________
| / 2 / / 2 2 \\ / 2 / / 2 2 \\| / 2 / / 2 2 \\ / 2 / / 2 2 \\
| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|
| \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------|| \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|
| \ 3 / \ 3 /| \ 3 / \ 3 /
x2 = I*|- --------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------
\ 2 2 / 2 2
$$x_{2} = i \left(- \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}$$
/ __________________________________________ __________________________________________ \ __________________________________________ __________________________________________
| / 2 / / 2 2 \\ / 2 / / 2 2 \\| / 2 / / 2 2 \\ / 2 / / 2 2 \\
| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|
| \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------|| \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|
| \ 3 / \ 3 /| \ 3 / \ 3 /
x3 = I*|- --------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------
\ 2 2 / 2 2
$$x_{3} = i \left(- \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}$$
x3 = i*(-((re(y)^2 - im(y)^2 + 8)^2 + 4*re(y)^2*im(y)^2)^(1/6)*sin(atan2(2*re(y)*im(y, re(y)^2 - im(y)^2 + 8)/3)/2 + sqrt(3)*((re(y)^2 - im(y)^2 + 8)^2 + 4*re(y)^2*im(y)^2)^(1/6)*cos(atan2(2*re(y)*im(y), re(y)^2 - im(y)^2 + 8)/3)/2) - sqrt(3)*((re(y)^2 - im(y)^2 + 8)^2 + 4*re(y)^2*im(y)^2)^(1/6)*sin(atan2(2*re(y)*im(y), re(y)^2 - im(y)^2 + 8)/3)/2 - ((re(y)^2 - im(y)^2 + 8)^2 + 4*re(y)^2*im(y)^2)^(1/6)*cos(atan2(2*re(y)*im(y), re(y)^2 - im(y)^2 + 8)/3)/2)
Suma y producto de raíces
[src]
suma
/ __________________________________________ __________________________________________ \ __________________________________________ __________________________________________ / __________________________________________ __________________________________________ \ __________________________________________ __________________________________________
| / 2 / / 2 2 \\ / 2 / / 2 2 \\| / 2 / / 2 2 \\ / 2 / / 2 2 \\ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| / 2 / / 2 2 \\ / 2 / / 2 2 \\
__________________________________________ __________________________________________ | 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| | 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ | \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------|| \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| | \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------|| \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|
6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| | \ 3 / \ 3 /| \ 3 / \ 3 / | \ 3 / \ 3 /| \ 3 / \ 3 /
\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| + I*\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| + I*|- --------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------- + I*|- --------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------
\ 3 / \ 3 / \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(\left(i \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)} + \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}\right) + \left(i \left(- \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right)\right) + \left(i \left(- \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right)$$
=
/ __________________________________________ __________________________________________ \ / __________________________________________ __________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| | 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| __________________________________________ | \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------|| | \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------|| / 2 / / 2 2 \\ | \ 3 / \ 3 /| | \ 3 / \ 3 /| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| I*|- --------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------| + I*|- --------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------| + I*\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \ 2 2 / \ 2 2 / \ 3 /
$$i \left(- \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) + i \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}$$
producto
/ / __________________________________________ __________________________________________ \ __________________________________________ __________________________________________ \ / / __________________________________________ __________________________________________ \ __________________________________________ __________________________________________ \
| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| / 2 / / 2 2 \\ / 2 / / 2 2 \\| | | / 2 / / 2 2 \\ / 2 / / 2 2 \\| / 2 / / 2 2 \\ / 2 / / 2 2 \\|
/ __________________________________________ __________________________________________ \ | | 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| | | 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| ___ 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/||
| / 2 / / 2 2 \\ / 2 / / 2 2 \\| | | \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------|| \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|| | | \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------|| \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| \/ 3 *\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------||
|6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| 6 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| | | \ 3 / \ 3 /| \ 3 / \ 3 /| | | \ 3 / \ 3 /| \ 3 / \ 3 /|
|\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| + I*\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------||*|I*|- --------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------|*|I*|- --------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------|
\ \ 3 / \ 3 // \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(i \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)} + \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}\right) \left(i \left(- \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) \left(i \left(- \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{2}\right)$$
=
/ / / / 2 2 \\\ \
| | 2*im(y)*re(y) |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| |
| 3*I*|---------------------------------------------- + sin|-----------------------------------------|| |
| / / 2 2 \\ | __________________________________________ \ 3 /| |
| |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| | / 2 | |
________________________________________________________________ | / / 2 2 \\ 3*cos|-----------------------------------------| / / 2 2 \\ 2 | / / 2 2 \ 2 2 | 2 |
/ 4 4 2 2 2 2 | 3|atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| 6 \ 3 / 3|atan2\2*im(y)*re(y), 8 + re (y) - im (y)/| 3*im (y) \\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) / 3*re (y) |
\/ 64 + im (y) + re (y) - 16*im (y) + 16*re (y) + 2*im (y)*re (y) *|cos |-----------------------------------------| + ---------------------------------------------- - ------------------------------------------------ - I*sin |-----------------------------------------| - ------------------------------------------------ + ----------------------------------------------------------------------------------------------------- + ------------------------------------------------|
| \ 3 / __________________________________________ 4 \ 3 / __________________________________________ 4 __________________________________________|
| / 2 / 2 / 2 |
| / / 2 2 \ 2 2 / / 2 2 \ 2 2 / / 2 2 \ 2 2 |
\ \/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) 4*\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) 4*\/ \8 + re (y) - im (y)/ + 4*im (y)*re (y) /
$$\sqrt{\left(\operatorname{re}{\left(y\right)}\right)^{4} + 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{4} - 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 64} \left(\frac{3 i \left(\sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)} + \frac{2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{\sqrt{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \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(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)} + \cos^{3}{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)} - \frac{3 \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{3} \right)}}{4} + \frac{3 \left(\operatorname{re}{\left(y\right)}\right)^{2}}{4 \sqrt{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}}} - \frac{3 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{4 \sqrt{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}}} + \frac{6}{\sqrt{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}}}\right)$$
sqrt(64 + im(y)^4 + re(y)^4 - 16*im(y)^2 + 16*re(y)^2 + 2*im(y)^2*re(y)^2)*(cos(atan2(2*im(y)*re(y), 8 + re(y)^2 - im(y)^2)/3)^3 + 6/sqrt((8 + re(y)^2 - im(y)^2)^2 + 4*im(y)^2*re(y)^2) - 3*cos(atan2(2*im(y)*re(y), 8 + re(y)^2 - im(y)^2)/3)/4 - i*sin(atan2(2*im(y)*re(y), 8 + re(y)^2 - im(y)^2)/3)^3 - 3*im(y)^2/(4*sqrt((8 + re(y)^2 - im(y)^2)^2 + 4*im(y)^2*re(y)^2)) + 3*i*(2*im(y)*re(y)/sqrt((8 + re(y)^2 - im(y)^2)^2 + 4*im(y)^2*re(y)^2) + sin(atan2(2*im(y)*re(y), 8 + re(y)^2 - im(y)^2)/3))/4 + 3*re(y)^2/(4*sqrt((8 + re(y)^2 - im(y)^2)^2 + 4*im(y)^2*re(y)^2)))