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Ecuación 3*x^2-x+2=0
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Ecuación 1-x/2=x/3
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Ecuación 3*x^2-9=0
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Ecuación 0,084:(6,2-x)=1,2
- Expresar {x} en función de y en la ecuación:
- 4*x-1*y=-3
- -11*x+9*y=-20
- -17*x+2*y=9
- 10*x-15*y=16
- Integral de d{x}:
- x^2*y
- Descomponer al cuadrado:
- x^2-x*y+y^2
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Expresiones idénticas
- x^ dos *y=x^ dos -x*y+y^ dos
- x al cuadrado multiplicar por y es igual a x al cuadrado menos x multiplicar por y más y al cuadrado
- x en el grado dos multiplicar por y es igual a x en el grado dos menos x multiplicar por y más y en el grado dos
- x2*y=x2-x*y+y2
- x²*y=x²-x*y+y²
- x en el grado 2*y=x en el grado 2-x*y+y en el grado 2
- x^2y=x^2-xy+y^2
- x2y=x2-xy+y2
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Expresiones semejantes
- x^2*y=x^2-x*y-y^2
- x^2*y=x^2+x*y+y^2
x^2*y=x^2-x*y+y^2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$x^{2} y = y^{2} + \left(x^{2} - x y\right)$$
en
$$x^{2} y + \left(- y^{2} + \left(- x^{2} + x y\right)\right) = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = x^{2} + x$$
$$c = - x^{2}$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = \frac{x^{2}}{2} + \frac{x}{2} - \frac{\sqrt{- 4 x^{2} + \left(x^{2} + x\right)^{2}}}{2}$$
$$y_{2} = \frac{x^{2}}{2} + \frac{x}{2} + \frac{\sqrt{- 4 x^{2} + \left(x^{2} + x\right)^{2}}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$x^{2} y = y^{2} + \left(x^{2} - x y\right)$$
en
$$x^{2} y + \left(- y^{2} + \left(- x^{2} + x y\right)\right) = 0$$
Es la ecuación de la forma
a*y^2 + b*y + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = x^{2} + x$$
$$c = - x^{2}$$
, entonces
D = b^2 - 4 * a * c =
(x + x^2)^2 - 4 * (-1) * (-x^2) = (x + x^2)^2 - 4*x^2
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = \frac{x^{2}}{2} + \frac{x}{2} - \frac{\sqrt{- 4 x^{2} + \left(x^{2} + x\right)^{2}}}{2}$$
$$y_{2} = \frac{x^{2}}{2} + \frac{x}{2} + \frac{\sqrt{- 4 x^{2} + \left(x^{2} + x\right)^{2}}}{2}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$x^{2} y = y^{2} + \left(x^{2} - x y\right)$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$- x^{2} y + x^{2} - x y + y^{2} = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - x^{2} - x$$
$$q = \frac{c}{a}$$
$$q = x^{2}$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = x^{2} + x$$
$$y_{1} y_{2} = x^{2}$$
$$x^{2} y = y^{2} + \left(x^{2} - x y\right)$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$- x^{2} y + x^{2} - x y + y^{2} = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - x^{2} - x$$
$$q = \frac{c}{a}$$
$$q = x^{2}$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = x^{2} + x$$
$$y_{1} y_{2} = x^{2}$$
Gráfica
Respuesta rápida
[src]
// ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \
|| / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | | | / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ |
|| 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| |
||- \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*re(x) |1 - \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*im(x)| |1 - \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*re(x) |- \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*im(x)
|\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / /
y1 = I*|---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------------------------------------
\ 2 2 / 2 2
$$y_{1} = i \left(\frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{im}{\left(x\right)}}{2}\right) - \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{im}{\left(x\right)}}{2} + \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{re}{\left(x\right)}}{2}$$
// ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \
|| / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | | | / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ |
||4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | |4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| |
||\/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*re(x) |1 + \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*im(x)| |1 + \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*re(x) |\/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*im(x)
|\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / /
y2 = I*|-------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------
\ 2 2 / 2 2
$$y_{2} = i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{im}{\left(x\right)}}{2}\right) - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{im}{\left(x\right)}}{2} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{re}{\left(x\right)}}{2}$$
y2 = i*((((2*re(x)*im(x) + 2*im(x))^2 + (re(x)^2 + 2*re(x) - im(x)^2 - 3)^2)^(1/4)*sin(atan2(2*re(x)*im(x) + 2*im(x, re(x)^2 + 2*re(x) - im(x)^2 - 3)/2) + im(x))*re(x)/2 + (((2*re(x)*im(x) + 2*im(x))^2 + (re(x)^2 + 2*re(x) - im(x)^2 - 3)^2)^(1/4)*cos(atan2(2*re(x)*im(x) + 2*im(x), re(x)^2 + 2*re(x) - im(x)^2 - 3)/2) + re(x) + 1)*im(x)/2) - (((2*re(x)*im(x) + 2*im(x))^2 + (re(x)^2 + 2*re(x) - im(x)^2 - 3)^2)^(1/4)*sin(atan2(2*re(x)*im(x) + 2*im(x), re(x)^2 + 2*re(x) - im(x)^2 - 3)/2) + im(x))*im(x)/2 + (((2*re(x)*im(x) + 2*im(x))^2 + (re(x)^2 + 2*re(x) - im(x)^2 - 3)^2)^(1/4)*cos(atan2(2*re(x)*im(x) + 2*im(x), re(x)^2 + 2*re(x) - im(x)^2 - 3)/2) + re(x) + 1)*re(x)/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 \\ | || 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | ||4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | |4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | ||- \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*re(x) |1 - \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*im(x)| |1 - \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*re(x) |- \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*im(x) ||\/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*re(x) |1 + \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*im(x)| |1 + \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*re(x) |\/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*im(x) |\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / / |\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / / I*|---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------ - ---------------------------------------------------------------------------------------------------------------------------------------------------------- + I*|-------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------------------- \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(i \left(\frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{im}{\left(x\right)}}{2}\right) - \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{im}{\left(x\right)}}{2} + \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{re}{\left(x\right)}}{2}\right) + \left(i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{im}{\left(x\right)}}{2}\right) - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{im}{\left(x\right)}}{2} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{re}{\left(x\right)}}{2}\right)$$
=
// ________________________________________________________________ \ / ________________________________________________________________ \ \ // ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \ / ________________________________________________________________ \ / ________________________________________________________________ \ || / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | | || / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | | | / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | ||4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | || 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | |4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | ||\/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*re(x) |1 + \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*im(x)| ||- \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*re(x) |1 - \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*im(x)| |1 + \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*re(x) |1 - \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*re(x) |\/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*im(x) |- \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*im(x) |\ \ 2 / / \ \ 2 / / | |\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / / \ \ 2 / / \ \ 2 / / I*|-------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| + I*|---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------- \ 2 2 / \ 2 2 / 2 2 2 2
$$i \left(\frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{im}{\left(x\right)}}{2}\right) + i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{im}{\left(x\right)}}{2}\right) - \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{im}{\left(x\right)}}{2} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{im}{\left(x\right)}}{2} + \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{re}{\left(x\right)}}{2}$$
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 \\ | | | || 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | ||4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | 4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | |4 / 2 / 2 2 \ |atan2\2*im(x) + 2*im(x)*re(x), -3 + re (x) - im (x) + 2*re(x)/| | | | ||- \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*re(x) |1 - \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*im(x)| |1 - \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*re(x) |- \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*im(x)| | ||\/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*re(x) |1 + \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*im(x)| |1 + \/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *cos|--------------------------------------------------------------| + re(x)|*re(x) |\/ (2*im(x) + 2*im(x)*re(x)) + \-3 + re (x) - im (x) + 2*re(x)/ *sin|--------------------------------------------------------------| + im(x)|*im(x)| | |\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / / | | |\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / / | |I*|---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------ - ----------------------------------------------------------------------------------------------------------------------------------------------------------|*|I*|-------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------| \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(i \left(\frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{im}{\left(x\right)}}{2}\right) - \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{im}{\left(x\right)}}{2} + \frac{\left(- \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{re}{\left(x\right)}}{2}\right) \left(i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{re}{\left(x\right)}}{2} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{im}{\left(x\right)}}{2}\right) - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{im}{\left(x\right)}\right) \operatorname{im}{\left(x\right)}}{2} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 3 \right)}}{2} \right)} + \operatorname{re}{\left(x\right)} + 1\right) \operatorname{re}{\left(x\right)}}{2}\right)$$
=
2 2 re (x) - im (x) + 2*I*im(x)*re(x)
$$\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 i \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2}$$
re(x)^2 - im(x)^2 + 2*i*im(x)*re(x)