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Expresiones idénticas
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- a*(x-2)*(x^2+4)+b*(x-2)*(x^2+4)+c*(x-2)*(x+2)=O
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Expresiones semejantes
- a*(x-2)*(x^2+4)+b*(x+2)*(x^2+4)+c*(x-2)*(x+2)=0
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- a*(x-2)*(x^2+4)+b*(x-2)*(x^2+4)+c*(x-2)*(x-2)=0
- a*(x-2)*(x^2-4)+b*(x-2)*(x^2+4)+c*(x-2)*(x+2)=0
- a*(x-2)*(x^2+4)+b*(x-2)*(x^2-4)+c*(x-2)*(x+2)=0
- a*(x-2)*(x^2+4)+b*(x-2)*(x^2+4)-c*(x-2)*(x+2)=0
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- a*(x-2)*(x^2+4)-b*(x-2)*(x^2+4)+c*(x-2)*(x+2)=0
a*(x-2)*(x^2+4)+b*(x-2)*(x^2+4)+c*(x-2)*(x+2)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
/ 2 \ / 2 \ a*(x - 2)*\x + 4/ + b*(x - 2)*\x + 4/ + c*(x - 2)*(x + 2) = 0
$$c \left(x - 2\right) \left(x + 2\right) + \left(a \left(x - 2\right) \left(x^{2} + 4\right) + b \left(x - 2\right) \left(x^{2} + 4\right)\right) = 0$$
Solución detallada
Tenemos la ecuación:
$$c \left(x - 2\right) \left(x + 2\right) + \left(a \left(x - 2\right) \left(x^{2} + 4\right) + b \left(x - 2\right) \left(x^{2} + 4\right)\right) = 0$$
cambiamos:
Saquemos el factor común fuera de paréntesis
$$\left(x - 2\right) \left(a x^{2} + 4 a + b x^{2} + 4 b + c x + 2 c\right) = 0$$
Ya que la parte derecha de la ecuación es igual a cero, la solución de la ecuación será, si por lo menos uno de los factores en la parte izquierda de la ecuación es igual a cero.
Obtenemos ecuaciones
$$4 a + 4 b + c x + 2 c + x^{2} \left(a + b\right) = 0$$
$$x - 2 = 0$$
resolvemos las ecuaciones obtenidas:
1.
$$4 a + 4 b + c x + 2 c + x^{2} \left(a + b\right) = 0$$
Abramos la expresión en la ecuación
$$4 a + 4 b + c x + 2 c + x^{2} \left(a + b\right) = 0$$
Obtenemos la ecuación cuadrática
$$a x^{2} + 4 a + b x^{2} + 4 b + c x + 2 c = 0$$
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
$$a = a + b$$
$$b = c$$
$$c = 4 a + 4 b + 2 c$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{- c + \sqrt{c^{2} - \left(4 a + 4 b\right) \left(4 a + 4 b + 2 c\right)}}{2 a + 2 b}$$
$$x_{2} = \frac{- c - \sqrt{c^{2} - \left(4 a + 4 b\right) \left(4 a + 4 b + 2 c\right)}}{2 a + 2 b}$$
2.
$$x - 2 = 0$$
Transportamos los términos libres (sin x)
del miembro izquierdo al derecho, obtenemos:
$$x = 2$$
Obtenemos la respuesta: x3 = 2
Entonces la respuesta definitiva es:
$$x_{1} = \frac{- c + \sqrt{c^{2} - \left(4 a + 4 b\right) \left(4 a + 4 b + 2 c\right)}}{2 a + 2 b}$$
$$x_{2} = \frac{- c - \sqrt{c^{2} - \left(4 a + 4 b\right) \left(4 a + 4 b + 2 c\right)}}{2 a + 2 b}$$
$$x_{3} = 2$$
$$c \left(x - 2\right) \left(x + 2\right) + \left(a \left(x - 2\right) \left(x^{2} + 4\right) + b \left(x - 2\right) \left(x^{2} + 4\right)\right) = 0$$
cambiamos:
Saquemos el factor común fuera de paréntesis
$$\left(x - 2\right) \left(a x^{2} + 4 a + b x^{2} + 4 b + c x + 2 c\right) = 0$$
Ya que la parte derecha de la ecuación es igual a cero, la solución de la ecuación será, si por lo menos uno de los factores en la parte izquierda de la ecuación es igual a cero.
Obtenemos ecuaciones
$$4 a + 4 b + c x + 2 c + x^{2} \left(a + b\right) = 0$$
$$x - 2 = 0$$
resolvemos las ecuaciones obtenidas:
1.
$$4 a + 4 b + c x + 2 c + x^{2} \left(a + b\right) = 0$$
Abramos la expresión en la ecuación
$$4 a + 4 b + c x + 2 c + x^{2} \left(a + b\right) = 0$$
Obtenemos la ecuación cuadrática
$$a x^{2} + 4 a + b x^{2} + 4 b + c x + 2 c = 0$$
Es la ecuación de la forma
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
$$a = a + b$$
$$b = c$$
$$c = 4 a + 4 b + 2 c$$
, entonces
D = b^2 - 4 * a * c =
(c)^2 - 4 * (a + b) * (2*c + 4*a + 4*b) = c^2 - (4*a + 4*b)*(2*c + 4*a + 4*b)
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{- c + \sqrt{c^{2} - \left(4 a + 4 b\right) \left(4 a + 4 b + 2 c\right)}}{2 a + 2 b}$$
$$x_{2} = \frac{- c - \sqrt{c^{2} - \left(4 a + 4 b\right) \left(4 a + 4 b + 2 c\right)}}{2 a + 2 b}$$
2.
$$x - 2 = 0$$
Transportamos los términos libres (sin x)
del miembro izquierdo al derecho, obtenemos:
$$x = 2$$
Obtenemos la respuesta: x3 = 2
Entonces la respuesta definitiva es:
$$x_{1} = \frac{- c + \sqrt{c^{2} - \left(4 a + 4 b\right) \left(4 a + 4 b + 2 c\right)}}{2 a + 2 b}$$
$$x_{2} = \frac{- c - \sqrt{c^{2} - \left(4 a + 4 b\right) \left(4 a + 4 b + 2 c\right)}}{2 a + 2 b}$$
$$x_{3} = 2$$
Gráfica
Respuesta rápida
[src]
x1 = 2
$$x_{1} = 2$$
/ / ____________________________________________________________________________________________________________________________________________________________________________________________________ \ / ____________________________________________________________________________________________________________________________________________________________________________________________________ \\ / ____________________________________________________________________________________________________________________________________________________________________________________________________ \ / ____________________________________________________________________________________________________________________________________________________________________________________________________ \
| | / 2 / / 2 2 2 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 2 2 2 2 \ |atan2\-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c), re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/| | | 4 / 2 / 2 2 2 2 2 2 \ |atan2\-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c), re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/| || | 4 / 2 / 2 2 2 2 2 2 \ |atan2\-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c), re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/| | | 4 / 2 / 2 2 2 2 2 2 \ |atan2\-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c), re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/| |
| (-2*im(a) - 2*im(b))*|- \/ (-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c)) + \re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + re(c)| (2*re(a) + 2*re(b))*|- \/ (-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c)) + \re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + im(c)|| (-2*im(a) - 2*im(b))*|- \/ (-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c)) + \re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + im(c)| (2*re(a) + 2*re(b))*|- \/ (-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c)) + \re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + re(c)|
| \ \ 2 / / \ \ 2 / /| \ \ 2 / / \ \ 2 / /
x2 = I*|- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ (2*im(a) + 2*im(b)) + (2*re(a) + 2*re(b)) (2*im(a) + 2*im(b)) + (2*re(a) + 2*re(b)) / (2*im(a) + 2*im(b)) + (2*re(a) + 2*re(b)) (2*im(a) + 2*im(b)) + (2*re(a) + 2*re(b))
$$x_{2} = \frac{\left(- \sqrt[4]{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)}\right)^{2} + \left(- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)},- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(c\right)}\right) \left(- 2 \operatorname{im}{\left(a\right)} - 2 \operatorname{im}{\left(b\right)}\right)}{\left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)^{2} + \left(2 \operatorname{im}{\left(a\right)} + 2 \operatorname{im}{\left(b\right)}\right)^{2}} - \frac{\left(- \sqrt[4]{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)}\right)^{2} + \left(- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)},- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(c\right)}\right) \left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)}{\left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)^{2} + \left(2 \operatorname{im}{\left(a\right)} + 2 \operatorname{im}{\left(b\right)}\right)^{2}} + i \left(- \frac{\left(- \sqrt[4]{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)}\right)^{2} + \left(- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)},- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(c\right)}\right) \left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)}{\left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)^{2} + \left(2 \operatorname{im}{\left(a\right)} + 2 \operatorname{im}{\left(b\right)}\right)^{2}} - \frac{\left(- \sqrt[4]{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)}\right)^{2} + \left(- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)},- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(c\right)}\right) \left(- 2 \operatorname{im}{\left(a\right)} - 2 \operatorname{im}{\left(b\right)}\right)}{\left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)^{2} + \left(2 \operatorname{im}{\left(a\right)} + 2 \operatorname{im}{\left(b\right)}\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 2 2 2 2 \\ |
| |4 / 2 / 2 2 2 2 2 2 \ |atan2\-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c), re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/| | |4 / 2 / 2 2 2 2 2 2 \ |atan2\-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c), re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/| || |4 / 2 / 2 2 2 2 2 2 \ |atan2\-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c), re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/| | |4 / 2 / 2 2 2 2 2 2 \ |atan2\-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c), re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/| |
| (-2*im(a) - 2*im(b))*|\/ (-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c)) + \re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + re(c)| (2*re(a) + 2*re(b))*|\/ (-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c)) + \re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + im(c)|| (-2*im(a) - 2*im(b))*|\/ (-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c)) + \re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + im(c)| (2*re(a) + 2*re(b))*|\/ (-32*im(a*b) - 8*im(a*c) - 8*im(b*c) - 32*im(a)*re(a) - 32*im(b)*re(b) + 2*im(c)*re(c)) + \re (c) - im (c) - 32*re(a*b) - 16*re (a) - 16*re (b) - 8*re(a*c) - 8*re(b*c) + 16*im (a) + 16*im (b)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + re(c)|
| \ \ 2 / / \ \ 2 / /| \ \ 2 / / \ \ 2 / /
x3 = I*|- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ (2*im(a) + 2*im(b)) + (2*re(a) + 2*re(b)) (2*im(a) + 2*im(b)) + (2*re(a) + 2*re(b)) / (2*im(a) + 2*im(b)) + (2*re(a) + 2*re(b)) (2*im(a) + 2*im(b)) + (2*re(a) + 2*re(b))
$$x_{3} = \frac{\left(\sqrt[4]{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)}\right)^{2} + \left(- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)},- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(c\right)}\right) \left(- 2 \operatorname{im}{\left(a\right)} - 2 \operatorname{im}{\left(b\right)}\right)}{\left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)^{2} + \left(2 \operatorname{im}{\left(a\right)} + 2 \operatorname{im}{\left(b\right)}\right)^{2}} - \frac{\left(\sqrt[4]{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)}\right)^{2} + \left(- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)},- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(c\right)}\right) \left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)}{\left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)^{2} + \left(2 \operatorname{im}{\left(a\right)} + 2 \operatorname{im}{\left(b\right)}\right)^{2}} + i \left(- \frac{\left(\sqrt[4]{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)}\right)^{2} + \left(- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)},- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(c\right)}\right) \left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)}{\left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)^{2} + \left(2 \operatorname{im}{\left(a\right)} + 2 \operatorname{im}{\left(b\right)}\right)^{2}} - \frac{\left(\sqrt[4]{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)}\right)^{2} + \left(- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 32 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} + 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} - 32 \operatorname{im}{\left(a b\right)} - 8 \operatorname{im}{\left(a c\right)} - 8 \operatorname{im}{\left(b c\right)},- 16 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 16 \left(\operatorname{re}{\left(b\right)}\right)^{2} + \left(\operatorname{re}{\left(c\right)}\right)^{2} - 32 \operatorname{re}{\left(a b\right)} - 8 \operatorname{re}{\left(a c\right)} - 8 \operatorname{re}{\left(b c\right)} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(b\right)}\right)^{2} - \left(\operatorname{im}{\left(c\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(c\right)}\right) \left(- 2 \operatorname{im}{\left(a\right)} - 2 \operatorname{im}{\left(b\right)}\right)}{\left(2 \operatorname{re}{\left(a\right)} + 2 \operatorname{re}{\left(b\right)}\right)^{2} + \left(2 \operatorname{im}{\left(a\right)} + 2 \operatorname{im}{\left(b\right)}\right)^{2}}\right)$$
x3 = (((-32*re(a)*im(a) - 32*re(b)*im(b) + 2*re(c)*im(c) - 32*im(a*b) - 8*im(a*c) - 8*im(b*c))^2 + (-16*re(a)^2 - 16*re(b)^2 + re(c)^2 - 32*re(a*b) - 8*re(a*c) - 8*re(b*c) + 16*im(a)^2 + 16*im(b)^2 - im(c)^2)^2)^(1/4)*sin(atan2(-32*re(a)*im(a) - 32*re(b)*im(b) + 2*re(c)*im(c) - 32*im(a*b) - 8*im(a*c) - 8*im(b*c, -16*re(a)^2 - 16*re(b)^2 + re(c)^2 - 32*re(a*b) - 8*re(a*c) - 8*re(b*c) + 16*im(a)^2 + 16*im(b)^2 - im(c)^2)/2) + im(c))*(-2*im(a) - 2*im(b))/((2*re(a) + 2*re(b))^2 + (2*im(a) + 2*im(b))^2) - (((-32*re(a)*im(a) - 32*re(b)*im(b) + 2*re(c)*im(c) - 32*im(a*b) - 8*im(a*c) - 8*im(b*c))^2 + (-16*re(a)^2 - 16*re(b)^2 + re(c)^2 - 32*re(a*b) - 8*re(a*c) - 8*re(b*c) + 16*im(a)^2 + 16*im(b)^2 - im(c)^2)^2)^(1/4)*cos(atan2(-32*re(a)*im(a) - 32*re(b)*im(b) + 2*re(c)*im(c) - 32*im(a*b) - 8*im(a*c) - 8*im(b*c), -16*re(a)^2 - 16*re(b)^2 + re(c)^2 - 32*re(a*b) - 8*re(a*c) - 8*re(b*c) + 16*im(a)^2 + 16*im(b)^2 - im(c)^2)/2) + re(c))*(2*re(a) + 2*re(b))/((2*re(a) + 2*re(b))^2 + (2*im(a) + 2*im(b))^2) + i*(-(((-32*re(a)*im(a) - 32*re(b)*im(b) + 2*re(c)*im(c) - 32*im(a*b) - 8*im(a*c) - 8*im(b*c))^2 + (-16*re(a)^2 - 16*re(b)^2 + re(c)^2 - 32*re(a*b) - 8*re(a*c) - 8*re(b*c) + 16*im(a)^2 + 16*im(b)^2 - im(c)^2)^2)^(1/4)*sin(atan2(-32*re(a)*im(a) - 32*re(b)*im(b) + 2*re(c)*im(c) - 32*im(a*b) - 8*im(a*c) - 8*im(b*c), -16*re(a)^2 - 16*re(b)^2 + re(c)^2 - 32*re(a*b) - 8*re(a*c) - 8*re(b*c) + 16*im(a)^2 + 16*im(b)^2 - im(c)^2)/2) + im(c))*(2*re(a) + 2*re(b))/((2*re(a) + 2*re(b))^2 + (2*im(a) + 2*im(b))^2) - (((-32*re(a)*im(a) - 32*re(b)*im(b) + 2*re(c)*im(c) - 32*im(a*b) - 8*im(a*c) - 8*im(b*c))^2 + (-16*re(a)^2 - 16*re(b)^2 + re(c)^2 - 32*re(a*b) - 8*re(a*c) - 8*re(b*c) + 16*im(a)^2 + 16*im(b)^2 - im(c)^2)^2)^(1/4)*cos(atan2(-32*re(a)*im(a) - 32*re(b)*im(b) + 2*re(c)*im(c) - 32*im(a*b) - 8*im(a*c) - 8*im(b*c), -16*re(a)^2 - 16*re(b)^2 + re(c)^2 - 32*re(a*b) - 8*re(a*c) - 8*re(b*c) + 16*im(a)^2 + 16*im(b)^2 - im(c)^2)/2) + re(c))*(-2*im(a) - 2*im(b))/((2*re(a) + 2*re(b))^2 + (2*im(a) + 2*im(b))^2)))