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- La ecuación:
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Ecuación (x+9)^2=36*x
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Ecuación a+120=2000/5
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Ecuación x^2+10*x+16=0
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Ecuación x^2+4*x-32=0
- Expresar {x} en función de y en la ecuación:
- -6*x-12*y=9
- 9*x-1*y=17
- 10*x-2*y=11
- 12*x-3*y=-2
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Expresiones idénticas
- ax²+3x- cuatro = cero
- ax² más 3x menos 4 es igual a 0
- ax² más 3x menos cuatro es igual a cero
- ax²+3x-4=O
-
Expresiones semejantes
- ax²+3x+4=0
- ax²-3x-4=0
ax²+3x-4=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$b = 3$$
$$c = -4$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{16 a + 9} - 3}{2 a}$$
$$x_{2} = \frac{- \sqrt{16 a + 9} - 3}{2 a}$$
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
True
$$b = 3$$
$$c = -4$$
, entonces
D = b^2 - 4 * a * c =
(3)^2 - 4 * (a) * (-4) = 9 + 16*a
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{16 a + 9} - 3}{2 a}$$
$$x_{2} = \frac{- \sqrt{16 a + 9} - 3}{2 a}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(a x^{2} + 3 x\right) - 4 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + 3 x - 4}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{3}{a}$$
$$q = \frac{c}{a}$$
$$q = - \frac{4}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{3}{a}$$
$$x_{1} x_{2} = - \frac{4}{a}$$
$$\left(a x^{2} + 3 x\right) - 4 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + 3 x - 4}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{3}{a}$$
$$q = \frac{c}{a}$$
$$q = - \frac{4}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{3}{a}$$
$$x_{1} x_{2} = - \frac{4}{a}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$a x^{2} + 3 x - 4 = 0$$
Коэффициент при x равен
$$a$$
entonces son posibles los casos para a :
$$a < 0$$
$$a = 0$$
Consideremos todos los casos con detalles:
Con
$$a < 0$$
la ecuación será
$$- x^{2} + 3 x - 4 = 0$$
su solución
Con
$$a = 0$$
la ecuación será
$$3 x - 4 = 0$$
su solución
$$x = \frac{4}{3}$$
$$a x^{2} + 3 x - 4 = 0$$
Коэффициент при x равен
$$a$$
entonces son posibles los casos para a :
$$a < 0$$
$$a = 0$$
Consideremos todos los casos con detalles:
Con
$$a < 0$$
la ecuación será
$$- x^{2} + 3 x - 4 = 0$$
su solución
Con
$$a = 0$$
la ecuación será
$$3 x - 4 = 0$$
su solución
$$x = \frac{4}{3}$$
Gráfica
Respuesta rápida
[src]
/ / ______________________________ \ ______________________________ \ / ______________________________ \ ______________________________
| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\
| |-3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*im(a) \/ (9 + 16*re(a)) + 256*im (a) *re(a)*sin|-----------------------------|| |-3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*re(a) \/ (9 + 16*re(a)) + 256*im (a) *im(a)*sin|-----------------------------|
| \ \ 2 // \ 2 /| \ \ 2 // \ 2 /
x1 = I*|- --------------------------------------------------------------------------------- + --------------------------------------------------------------------------| + --------------------------------------------------------------------------------- + --------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$x_{1} = \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} - 3\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} - 3\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
// ______________________________ \ ______________________________ \ / ______________________________ \ ______________________________
|| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\
||3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*im(a) \/ (9 + 16*re(a)) + 256*im (a) *re(a)*sin|-----------------------------|| |3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*re(a) \/ (9 + 16*re(a)) + 256*im (a) *im(a)*sin|-----------------------------|
|\ \ 2 // \ 2 /| \ \ 2 // \ 2 /
x2 = I*|-------------------------------------------------------------------------------- - --------------------------------------------------------------------------| - -------------------------------------------------------------------------------- - --------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$x_{2} = - \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} + 3\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} + 3\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
x2 = -(((16*re(a) + 9)^2 + 256*im(a)^2)^(1/4)*cos(atan2(16*im(a, 16*re(a) + 9)/2) + 3)*re(a)/(2*(re(a)^2 + im(a)^2)) + i*((((16*re(a) + 9)^2 + 256*im(a)^2)^(1/4)*cos(atan2(16*im(a), 16*re(a) + 9)/2) + 3)*im(a)/(2*(re(a)^2 + im(a)^2)) - ((16*re(a) + 9)^2 + 256*im(a)^2)^(1/4)*sin(atan2(16*im(a), 16*re(a) + 9)/2)*re(a)/(2*(re(a)^2 + im(a)^2))) - ((16*re(a) + 9)^2 + 256*im(a)^2)^(1/4)*sin(atan2(16*im(a), 16*re(a) + 9)/2)*im(a)/(2*(re(a)^2 + im(a)^2)))
Suma y producto de raíces
[src]
suma
/ / ______________________________ \ ______________________________ \ / ______________________________ \ ______________________________ // ______________________________ \ ______________________________ \ / ______________________________ \ ______________________________ | | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\ || 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\ | |-3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*im(a) \/ (9 + 16*re(a)) + 256*im (a) *re(a)*sin|-----------------------------|| |-3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*re(a) \/ (9 + 16*re(a)) + 256*im (a) *im(a)*sin|-----------------------------| ||3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*im(a) \/ (9 + 16*re(a)) + 256*im (a) *re(a)*sin|-----------------------------|| |3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*re(a) \/ (9 + 16*re(a)) + 256*im (a) *im(a)*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 (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$\left(\frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} - 3\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} - 3\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \left(- \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} + 3\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} + 3\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
=
// ______________________________ \ ______________________________ \ / / ______________________________ \ ______________________________ \ / ______________________________ \ / ______________________________ \ || 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| ||3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*im(a) \/ (9 + 16*re(a)) + 256*im (a) *re(a)*sin|-----------------------------|| | |-3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*im(a) \/ (9 + 16*re(a)) + 256*im (a) *re(a)*sin|-----------------------------|| |-3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*re(a) |3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*re(a) |\ \ 2 // \ 2 /| | \ \ 2 // \ 2 /| \ \ 2 // \ \ 2 // I*|-------------------------------------------------------------------------------- - --------------------------------------------------------------------------| + I*|- --------------------------------------------------------------------------------- + --------------------------------------------------------------------------| + --------------------------------------------------------------------------------- - -------------------------------------------------------------------------------- | / 2 2 \ / 2 2 \ | | / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$\frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} - 3\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} + 3\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} - 3\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + i \left(\frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} + 3\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
producto
/ / / ______________________________ \ ______________________________ \ / ______________________________ \ ______________________________ \ / // ______________________________ \ ______________________________ \ / ______________________________ \ ______________________________ \ | | | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | || 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| 4 / 2 2 /atan2(16*im(a), 9 + 16*re(a))\| | | |-3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*im(a) \/ (9 + 16*re(a)) + 256*im (a) *re(a)*sin|-----------------------------|| |-3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*re(a) \/ (9 + 16*re(a)) + 256*im (a) *im(a)*sin|-----------------------------|| | ||3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*im(a) \/ (9 + 16*re(a)) + 256*im (a) *re(a)*sin|-----------------------------|| |3 + \/ (9 + 16*re(a)) + 256*im (a) *cos|-----------------------------||*re(a) \/ (9 + 16*re(a)) + 256*im (a) *im(a)*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 (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / \ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ /
$$\left(\frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} - 3\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} - 3\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) \left(- \frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} + 3\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} + 3\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(a\right)} + 9\right)^{2} + 256 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{im}{\left(a\right)},16 \operatorname{re}{\left(a\right)} + 9 \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
=
-4*re(a) + 4*I*im(a)
--------------------
2 2
im (a) + re (a)
$$\frac{- 4 \operatorname{re}{\left(a\right)} + 4 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
(-4*re(a) + 4*i*im(a))/(im(a)^2 + re(a)^2)