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Expresiones semejantes
- (x-2)^2+(y+1)^2-z^2=4
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(x-2)^2+(y+1)^2+z^2=4 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 2 (x - 2) + (y + 1) + z = 4
$$z^{2} + \left(\left(x - 2\right)^{2} + \left(y + 1\right)^{2}\right) = 4$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$z^{2} + \left(\left(x - 2\right)^{2} + \left(y + 1\right)^{2}\right) = 4$$
en
$$\left(z^{2} + \left(\left(x - 2\right)^{2} + \left(y + 1\right)^{2}\right)\right) - 4 = 0$$
Abramos la expresión en la ecuación
$$\left(z^{2} + \left(\left(x - 2\right)^{2} + \left(y + 1\right)^{2}\right)\right) - 4 = 0$$
Obtenemos la ecuación cuadrática
$$x^{2} - 4 x + y^{2} + 2 y + z^{2} + 1 = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = -4$$
$$c = y^{2} + 2 y + z^{2} + 1$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{- 4 y^{2} - 8 y - 4 z^{2} + 12}}{2} + 2$$
$$x_{2} = 2 - \frac{\sqrt{- 4 y^{2} - 8 y - 4 z^{2} + 12}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$z^{2} + \left(\left(x - 2\right)^{2} + \left(y + 1\right)^{2}\right) = 4$$
en
$$\left(z^{2} + \left(\left(x - 2\right)^{2} + \left(y + 1\right)^{2}\right)\right) - 4 = 0$$
Abramos la expresión en la ecuación
$$\left(z^{2} + \left(\left(x - 2\right)^{2} + \left(y + 1\right)^{2}\right)\right) - 4 = 0$$
Obtenemos la ecuación cuadrática
$$x^{2} - 4 x + y^{2} + 2 y + z^{2} + 1 = 0$$
Es la ecuación de la forma
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = -4$$
$$c = y^{2} + 2 y + z^{2} + 1$$
, entonces
D = b^2 - 4 * a * c =
(-4)^2 - 4 * (1) * (1 + y^2 + z^2 + 2*y) = 12 - 8*y - 4*y^2 - 4*z^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{- 4 y^{2} - 8 y - 4 z^{2} + 12}}{2} + 2$$
$$x_{2} = 2 - \frac{\sqrt{- 4 y^{2} - 8 y - 4 z^{2} + 12}}{2}$$
Gráfica
Respuesta rápida
[src]
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/ 2 / / 2 2 2 2 \\ / 2 / / 2 2 2 2 \\
4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/|
x1 = 2 - \/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *cos|------------------------------------------------------------------------------------------------| - I*\/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *sin|------------------------------------------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + 2$$
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/ 2 / / 2 2 2 2 \\ / 2 / / 2 2 2 2 \\
4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/|
x2 = 2 + \/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *cos|------------------------------------------------------------------------------------------------| + I*\/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *sin|------------------------------------------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + 2$$
x2 = i*((-2*re(y)*im(y) - 2*re(z)*im(z) - 2*im(y))^2 + (-re(y)^2 - 2*re(y) - re(z)^2 + im(y)^2 + im(z)^2 + 3)^2)^(1/4)*sin(atan2(-2*re(y)*im(y) - 2*re(z)*im(z) - 2*im(y, -re(y)^2 - 2*re(y) - re(z)^2 + im(y)^2 + im(z)^2 + 3)/2) + ((-2*re(y)*im(y) - 2*re(z)*im(z) - 2*im(y))^2 + (-re(y)^2 - 2*re(y) - re(z)^2 + im(y)^2 + im(z)^2 + 3)^2)^(1/4)*cos(atan2(-2*re(y)*im(y) - 2*re(z)*im(z) - 2*im(y), -re(y)^2 - 2*re(y) - re(z)^2 + im(y)^2 + im(z)^2 + 3)/2) + 2)
Suma y producto de raíces
[src]
suma
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/ 2 / / 2 2 2 2 \\ / 2 / / 2 2 2 2 \\ / 2 / / 2 2 2 2 \\ / 2 / / 2 2 2 2 \\
4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/|
2 - \/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *cos|------------------------------------------------------------------------------------------------| - I*\/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *sin|------------------------------------------------------------------------------------------------| + 2 + \/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *cos|------------------------------------------------------------------------------------------------| + I*\/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *sin|------------------------------------------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + 2\right) + \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + 2\right)$$
=
4
$$4$$
producto
/ __________________________________________________________________________________________________ __________________________________________________________________________________________________ \ / __________________________________________________________________________________________________ __________________________________________________________________________________________________ \ | / 2 / / 2 2 2 2 \\ / 2 / / 2 2 2 2 \\| | / 2 / / 2 2 2 2 \\ / 2 / / 2 2 2 2 \\| | 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/|| | 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z), 3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/|| |2 - \/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *cos|------------------------------------------------------------------------------------------------| - I*\/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *sin|------------------------------------------------------------------------------------------------||*|2 + \/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *cos|------------------------------------------------------------------------------------------------| + I*\/ (-2*im(y) - 2*im(y)*re(y) - 2*im(z)*re(z)) + \3 + im (y) + im (z) - re (y) - re (z) - 2*re(y)/ *sin|------------------------------------------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + 2\right) \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{re}{\left(z\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(z\right)}\right)^{2} + 3 \right)}}{2} \right)} + 2\right)$$
=
2 2 2 2 1 + re (y) + re (z) - im (y) - im (z) + 2*re(y) + 2*I*im(y) + 2*I*im(y)*re(y) + 2*I*im(z)*re(z)
$$\left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{re}{\left(z\right)}\right)^{2} + 2 i \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2 i \operatorname{im}{\left(y\right)} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 1$$
1 + re(y)^2 + re(z)^2 - im(y)^2 - im(z)^2 + 2*re(y) + 2*i*im(y) + 2*i*im(y)*re(y) + 2*i*im(z)*re(z)