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Ecuación 3*x^2-x+2=0
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Ecuación 3*x^2-9=0
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Ecuación 0,084:(6,2-x)=1,2
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Expresiones idénticas
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Expresiones semejantes
- ax^2-2x^2+2ax+a-2-x=0
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- ax^2+2x^2-2ax+a-2-x=0
- ax^2+2x^2+2ax-a-2-x=0
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Expresiones con funciones
- ax
- ax+b-b=c-b
- ax-b/a+b+bx+a/a-b=a^2+b^2/a^2-b^2
- ax−8=3x
- ax+b=x
- ax-3x=a^2-9
ax^2+2x^2+2ax+a-2-x=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 a*x + 2*x + 2*a*x + a - 2 - x = 0
$$- x + \left(\left(a + \left(2 a x + \left(a x^{2} + 2 x^{2}\right)\right)\right) - 2\right) = 0$$
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
$$b = 2 a - 1$$
$$c = a - 2$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{- 2 a + \sqrt{- \left(a - 2\right) \left(4 a + 8\right) + \left(2 a - 1\right)^{2}} + 1}{2 a + 4}$$
$$x_{2} = \frac{- 2 a - \sqrt{- \left(a - 2\right) \left(4 a + 8\right) + \left(2 a - 1\right)^{2}} + 1}{2 a + 4}$$
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
False
$$b = 2 a - 1$$
$$c = a - 2$$
, entonces
D = b^2 - 4 * a * c =
(-1 + 2*a)^2 - 4 * (2 + a) * (-2 + a) = (-1 + 2*a)^2 - (-2 + a)*(8 + 4*a)
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{- 2 a + \sqrt{- \left(a - 2\right) \left(4 a + 8\right) + \left(2 a - 1\right)^{2}} + 1}{2 a + 4}$$
$$x_{2} = \frac{- 2 a - \sqrt{- \left(a - 2\right) \left(4 a + 8\right) + \left(2 a - 1\right)^{2}} + 1}{2 a + 4}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$a x^{2} + 2 a x + a + 2 x^{2} - x - 2 = 0$$
Коэффициент при x равен
$$a + 2$$
entonces son posibles los casos para a :
$$a < -2$$
$$a = -2$$
Consideremos todos los casos con detalles:
Con
$$a < -2$$
la ecuación será
$$- x^{2} - 7 x - 5 = 0$$
su solución
$$x = - \frac{7}{2} - \frac{\sqrt{29}}{2}$$
$$x = - \frac{7}{2} + \frac{\sqrt{29}}{2}$$
Con
$$a = -2$$
la ecuación será
$$- 5 x - 4 = 0$$
su solución
$$x = - \frac{4}{5}$$
$$a x^{2} + 2 a x + a + 2 x^{2} - x - 2 = 0$$
Коэффициент при x равен
$$a + 2$$
entonces son posibles los casos para a :
$$a < -2$$
$$a = -2$$
Consideremos todos los casos con detalles:
Con
$$a < -2$$
la ecuación será
$$- x^{2} - 7 x - 5 = 0$$
su solución
$$x = - \frac{7}{2} - \frac{\sqrt{29}}{2}$$
$$x = - \frac{7}{2} + \frac{\sqrt{29}}{2}$$
Con
$$a = -2$$
la ecuación será
$$- 5 x - 4 = 0$$
su solución
$$x = - \frac{4}{5}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$- x + \left(\left(a + \left(2 a x + \left(a x^{2} + 2 x^{2}\right)\right)\right) - 2\right) = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + 2 a x + a + 2 x^{2} - x - 2}{a + 2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{2 a - 1}{a + 2}$$
$$q = \frac{c}{a}$$
$$q = \frac{a - 2}{a + 2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2 a - 1}{a + 2}$$
$$x_{1} x_{2} = \frac{a - 2}{a + 2}$$
$$- x + \left(\left(a + \left(2 a x + \left(a x^{2} + 2 x^{2}\right)\right)\right) - 2\right) = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + 2 a x + a + 2 x^{2} - x - 2}{a + 2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{2 a - 1}{a + 2}$$
$$q = \frac{c}{a}$$
$$q = \frac{a - 2}{a + 2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2 a - 1}{a + 2}$$
$$x_{1} x_{2} = \frac{a - 2}{a + 2}$$
Gráfica
Respuesta rápida
[src]
/ / _____________________________ \ / _____________________________ \ \ / _____________________________ \ / _____________________________ \
| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\|
|(2 + re(a))*|-2*im(a) - \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------|| |1 - 2*re(a) - \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a)| (2 + re(a))*|1 - 2*re(a) - \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------|| |-2*im(a) - \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------||*im(a)
| \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 //
x1 = I*|-------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/
$$x_{1} = \frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
/ / _____________________________ \ / _____________________________ \ \ / _____________________________ \ / _____________________________ \
| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\|
|(2 + re(a))*|-2*im(a) + \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------|| |1 - 2*re(a) + \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a)| (2 + re(a))*|1 - 2*re(a) + \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------|| |-2*im(a) + \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------||*im(a)
| \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 //
x2 = I*|-------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/
$$x_{2} = \frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
x2 = (((17 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*sin(atan2(-4*im(a, 17 - 4*re(a))/2) - 2*im(a))*im(a)/(2*((re(a) + 2)^2 + im(a)^2)) + i*((((17 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*sin(atan2(-4*im(a), 17 - 4*re(a))/2) - 2*im(a))*(re(a) + 2)/(2*((re(a) + 2)^2 + im(a)^2)) - (((17 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*cos(atan2(-4*im(a), 17 - 4*re(a))/2) - 2*re(a) + 1)*im(a)/(2*((re(a) + 2)^2 + im(a)^2))) + (re(a) + 2)*(((17 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*cos(atan2(-4*im(a), 17 - 4*re(a))/2) - 2*re(a) + 1)/(2*((re(a) + 2)^2 + im(a)^2)))
Suma y producto de raíces
[src]
suma
/ / _____________________________ \ / _____________________________ \ \ / _____________________________ \ / _____________________________ \ / / _____________________________ \ / _____________________________ \ \ / _____________________________ \ / _____________________________ \ | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| |(2 + re(a))*|-2*im(a) - \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------|| |1 - 2*re(a) - \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a)| (2 + re(a))*|1 - 2*re(a) - \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------|| |-2*im(a) - \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------||*im(a) |(2 + re(a))*|-2*im(a) + \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------|| |1 - 2*re(a) + \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a)| (2 + re(a))*|1 - 2*re(a) + \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------|| |-2*im(a) + \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------||*im(a) | \ \ 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*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ \ 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/
$$\left(\frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \left(\frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
=
/ / _____________________________ \ / _____________________________ \ \ / / _____________________________ \ / _____________________________ \ \ / _____________________________ \ / _____________________________ \ / _____________________________ \ / _____________________________ \ | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| |(2 + re(a))*|-2*im(a) + \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------|| |1 - 2*re(a) + \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a)| |(2 + re(a))*|-2*im(a) - \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------|| |1 - 2*re(a) - \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a)| (2 + re(a))*|1 - 2*re(a) + \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------|| (2 + re(a))*|1 - 2*re(a) - \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------|| |-2*im(a) + \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------||*im(a) |-2*im(a) - \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------||*im(a) | \ \ 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*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / \ 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/
$$\frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + i \left(\frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
producto
/ / / _____________________________ \ / _____________________________ \ \ / _____________________________ \ / _____________________________ \ \ / / / _____________________________ \ / _____________________________ \ \ / _____________________________ \ / _____________________________ \ \ | | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 17 - 4*re(a))\| | | |(2 + re(a))*|-2*im(a) - \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------|| |1 - 2*re(a) - \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a)| (2 + re(a))*|1 - 2*re(a) - \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------|| |-2*im(a) - \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------||*im(a)| | |(2 + re(a))*|-2*im(a) + \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------|| |1 - 2*re(a) + \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a)| (2 + re(a))*|1 - 2*re(a) + \/ (17 - 4*re(a)) + 16*im (a) *cos|-----------------------------|| |-2*im(a) + \/ (17 - 4*re(a)) + 16*im (a) *sin|-----------------------------||*im(a)| | | \ \ 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*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / \ \ 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ / 2*\(2 + re(a)) + im (a)/ 2*\(2 + re(a)) + im (a)/ /
$$\left(\frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(- \sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) \left(\frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(\sqrt[4]{\left(17 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},17 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 2 \operatorname{re}{\left(a\right)} + 1\right)}{2 \left(\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
=
2 2
-4 + im (a) + re (a) + 4*I*im(a)
--------------------------------
2 2
4 + im (a) + re (a) + 4*re(a)
$$\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 4 i \operatorname{im}{\left(a\right)} - 4}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 4}$$
(-4 + im(a)^2 + re(a)^2 + 4*i*im(a))/(4 + im(a)^2 + re(a)^2 + 4*re(a))