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La ecuación:
Ecuación 3*x^2-x+2=0
Ecuación x^2=-74
Ecuación z^2+z+1=0
Ecuación 90+x-4*10=60
Expresar {x} en función de y en la ecuación:
9*x+14*y=-10
-4*x-8*y=-20
-12*x-10*y=-16
10*x-15*y=16
Integral de d{x}:
x^2*y
Descomponer al cuadrado:
x^2-x*y+y^2
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
Expresiones semejantes
x^2*y=x^2+x*y+y^2
x^2*y=x^2-x*y-y^2

x^2y=x^2-xy+y^2 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

 2      2          2
x *y = x  - x*y + y

x^{2} y = y^{2} + \left(x^{2} - x y\right)

Simplificar

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)

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)

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)

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)

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