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Ecuación -8x-17=3x-105
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Xyy+1=y la ecuación
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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
$$y x y + 1 = y$$
en
$$- y + \left(y x y + 1\right) = 0$$
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
$$a = x$$
$$b = -1$$
$$c = 1$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = \frac{\sqrt{1 - 4 x} + 1}{2 x}$$
$$y_{2} = \frac{1 - \sqrt{1 - 4 x}}{2 x}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$y x y + 1 = y$$
en
$$- y + \left(y x y + 1\right) = 0$$
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
$$a = x$$
$$b = -1$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (x) * (1) = 1 - 4*x
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = \frac{\sqrt{1 - 4 x} + 1}{2 x}$$
$$y_{2} = \frac{1 - \sqrt{1 - 4 x}}{2 x}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$x y^{2} + 1 = y$$
Коэффициент при y равен
$$x$$
entonces son posibles los casos para x :
$$x < 0$$
$$x = 0$$
Consideremos todos los casos con detalles:
Con
$$x < 0$$
la ecuación será
$$- y^{2} - y + 1 = 0$$
su solución
$$y = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$y = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Con
$$x = 0$$
la ecuación será
$$1 - y = 0$$
su solución
$$y = 1$$
$$x y^{2} + 1 = y$$
Коэффициент при y равен
$$x$$
entonces son posibles los casos para x :
$$x < 0$$
$$x = 0$$
Consideremos todos los casos con detalles:
Con
$$x < 0$$
la ecuación será
$$- y^{2} - y + 1 = 0$$
su solución
$$y = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$y = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Con
$$x = 0$$
la ecuación será
$$1 - y = 0$$
su solución
$$y = 1$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$y x y + 1 = y$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$\frac{x y^{2} - y + 1}{x} = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{1}{x}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{x}$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = \frac{1}{x}$$
$$y_{1} y_{2} = \frac{1}{x}$$
$$y x y + 1 = y$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$\frac{x y^{2} - y + 1}{x} = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{1}{x}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{x}$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = \frac{1}{x}$$
$$y_{1} y_{2} = \frac{1}{x}$$
Gráfica
Respuesta rápida
[src]
/ / ____________________________ \ ____________________________ \ / ____________________________ \ ____________________________
| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\
| |1 - \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*im(x) \/ (1 - 4*re(x)) + 16*im (x) *re(x)*sin|----------------------------|| |1 - \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*re(x) \/ (1 - 4*re(x)) + 16*im (x) *im(x)*sin|----------------------------|
| \ \ 2 // \ 2 /| \ \ 2 // \ 2 /
y1 = I*|- ----------------------------------------------------------------------------- - -----------------------------------------------------------------------| + ----------------------------------------------------------------------------- - -----------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / 2*\im (x) + re (x)/ 2*\im (x) + re (x)/
$$y_{1} = \frac{\left(- \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}$$
/ / ____________________________ \ ____________________________ \ / ____________________________ \ ____________________________
| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\
| |1 + \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*im(x) \/ (1 - 4*re(x)) + 16*im (x) *re(x)*sin|----------------------------|| |1 + \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*re(x) \/ (1 - 4*re(x)) + 16*im (x) *im(x)*sin|----------------------------|
| \ \ 2 // \ 2 /| \ \ 2 // \ 2 /
y2 = I*|- ----------------------------------------------------------------------------- + -----------------------------------------------------------------------| + ----------------------------------------------------------------------------- + -----------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / 2*\im (x) + re (x)/ 2*\im (x) + re (x)/
$$y_{2} = \frac{\left(\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}$$
y2 = (((1 - 4*re(x))^2 + 16*im(x)^2)^(1/4)*cos(atan2(-4*im(x, 1 - 4*re(x))/2) + 1)*re(x)/(2*(re(x)^2 + im(x)^2)) + i*(-(((1 - 4*re(x))^2 + 16*im(x)^2)^(1/4)*cos(atan2(-4*im(x), 1 - 4*re(x))/2) + 1)*im(x)/(2*(re(x)^2 + im(x)^2)) + ((1 - 4*re(x))^2 + 16*im(x)^2)^(1/4)*sin(atan2(-4*im(x), 1 - 4*re(x))/2)*re(x)/(2*(re(x)^2 + im(x)^2))) + ((1 - 4*re(x))^2 + 16*im(x)^2)^(1/4)*sin(atan2(-4*im(x), 1 - 4*re(x))/2)*im(x)/(2*(re(x)^2 + im(x)^2)))
Suma y producto de raíces
[src]
suma
/ / ____________________________ \ ____________________________ \ / ____________________________ \ ____________________________ / / ____________________________ \ ____________________________ \ / ____________________________ \ ____________________________ | | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\ | | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\ | |1 - \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*im(x) \/ (1 - 4*re(x)) + 16*im (x) *re(x)*sin|----------------------------|| |1 - \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*re(x) \/ (1 - 4*re(x)) + 16*im (x) *im(x)*sin|----------------------------| | |1 + \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*im(x) \/ (1 - 4*re(x)) + 16*im (x) *re(x)*sin|----------------------------|| |1 + \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*re(x) \/ (1 - 4*re(x)) + 16*im (x) *im(x)*sin|----------------------------| | \ \ 2 // \ 2 /| \ \ 2 // \ 2 / | \ \ 2 // \ 2 /| \ \ 2 // \ 2 / I*|- ----------------------------------------------------------------------------- - -----------------------------------------------------------------------| + ----------------------------------------------------------------------------- - ----------------------------------------------------------------------- + I*|- ----------------------------------------------------------------------------- + -----------------------------------------------------------------------| + ----------------------------------------------------------------------------- + ----------------------------------------------------------------------- | / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \ \ 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ \ 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / 2*\im (x) + re (x)/ 2*\im (x) + re (x)/
$$\left(\frac{\left(- \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) + \left(\frac{\left(\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right)$$
=
/ / ____________________________ \ ____________________________ \ / / ____________________________ \ ____________________________ \ / ____________________________ \ / ____________________________ \ | | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | |1 + \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*im(x) \/ (1 - 4*re(x)) + 16*im (x) *re(x)*sin|----------------------------|| | |1 - \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*im(x) \/ (1 - 4*re(x)) + 16*im (x) *re(x)*sin|----------------------------|| |1 + \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*re(x) |1 - \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*re(x) | \ \ 2 // \ 2 /| | \ \ 2 // \ 2 /| \ \ 2 // \ \ 2 // I*|- ----------------------------------------------------------------------------- + -----------------------------------------------------------------------| + I*|- ----------------------------------------------------------------------------- - -----------------------------------------------------------------------| + ----------------------------------------------------------------------------- + ----------------------------------------------------------------------------- | / 2 2 \ / 2 2 \ | | / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \ \ 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / \ 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / 2*\im (x) + re (x)/ 2*\im (x) + re (x)/
$$\frac{\left(- \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) + i \left(- \frac{\left(\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right)$$
producto
/ / / ____________________________ \ ____________________________ \ / ____________________________ \ ____________________________ \ / / / ____________________________ \ ____________________________ \ / ____________________________ \ ____________________________ \ | | | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | | | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| 4 / 2 2 /atan2(-4*im(x), 1 - 4*re(x))\| | | |1 - \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*im(x) \/ (1 - 4*re(x)) + 16*im (x) *re(x)*sin|----------------------------|| |1 - \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*re(x) \/ (1 - 4*re(x)) + 16*im (x) *im(x)*sin|----------------------------|| | | |1 + \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*im(x) \/ (1 - 4*re(x)) + 16*im (x) *re(x)*sin|----------------------------|| |1 + \/ (1 - 4*re(x)) + 16*im (x) *cos|----------------------------||*re(x) \/ (1 - 4*re(x)) + 16*im (x) *im(x)*sin|----------------------------|| | | \ \ 2 // \ 2 /| \ \ 2 // \ 2 /| | | \ \ 2 // \ 2 /| \ \ 2 // \ 2 /| |I*|- ----------------------------------------------------------------------------- - -----------------------------------------------------------------------| + ----------------------------------------------------------------------------- - -----------------------------------------------------------------------|*|I*|- ----------------------------------------------------------------------------- + -----------------------------------------------------------------------| + ----------------------------------------------------------------------------- + -----------------------------------------------------------------------| | | / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \ | | | / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \ | \ \ 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / \ \ 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ / 2*\im (x) + re (x)/ 2*\im (x) + re (x)/ /
$$\left(\frac{\left(- \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) \left(\frac{\left(\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{re}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(x\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(x\right)},1 - 4 \operatorname{re}{\left(x\right)} \right)}}{2} \right)} \operatorname{im}{\left(x\right)}}{2 \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)}\right)$$
=
-I*im(x) + re(x) ---------------- 2 2 im (x) + re (x)
$$\frac{\operatorname{re}{\left(x\right)} - i \operatorname{im}{\left(x\right)}}{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}$$
(-i*im(x) + re(x))/(im(x)^2 + re(x)^2)