Sr Examen
Otras calculadoras
-
¿Cómo usar?
- La ecuación:
-
Ecuación 49x^3+14x^2+x=0
-
Ecuación (x-3)/(x+1)=0
-
Ecuación 15x=0,15
-
Ecuación (x-2)/(x-3)=2
- Expresar {x} en función de y en la ecuación:
- -16*x+3*y=17
- 18*x-14*y=-2
- 18*x+19*y=10
- 11*x-6*y=5
-
Expresiones idénticas
- (a+ dos)x^ dos - dos (a+ tres)x+a+ cinco = cero
- (a más 2)x al cuadrado menos 2(a más 3)x más a más 5 es igual a 0
- (a más dos)x en el grado dos menos dos (a más tres)x más a más cinco es igual a cero
- (a+2)x2-2(a+3)x+a+5=0
- a+2x2-2a+3x+a+5=0
- (a+2)x²-2(a+3)x+a+5=0
- (a+2)x en el grado 2-2(a+3)x+a+5=0
- a+2x^2-2a+3x+a+5=0
- (a+2)x^2-2(a+3)x+a+5=O
-
Expresiones semejantes
- (a-2)x^2-2(a+3)x+a+5=0
- (a+2)x^2-2(a+3)x+a-5=0
- (a+2)x^2+2(a+3)x+a+5=0
- (a+2)x^2-2(a-3)x+a+5=0
- (a+2)x^2-2(a+3)x-a+5=0
(a+2)x^2-2(a+3)x+a+5=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 (a + 2)*x - 2*(a + 3)*x + a + 5 = 0
$$\left(a + \left(x^{2} \left(a + 2\right) - x 2 \left(a + 3\right)\right)\right) + 5 = 0$$
Solución detallada
Abramos la expresión en la ecuación
$$\left(a + \left(x^{2} \left(a + 2\right) - x 2 \left(a + 3\right)\right)\right) + 5 = 0$$
Obtenemos la ecuación cuadrática
$$a x^{2} - 2 a x + a + 2 x^{2} - 6 x + 5 = 0$$
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
$$b = - 2 a - 6$$
$$c = a + 5$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{2 a + \sqrt{\left(- 2 a - 6\right)^{2} - \left(a + 5\right) \left(4 a + 8\right)} + 6}{2 a + 4}$$
$$x_{2} = \frac{2 a - \sqrt{\left(- 2 a - 6\right)^{2} - \left(a + 5\right) \left(4 a + 8\right)} + 6}{2 a + 4}$$
$$\left(a + \left(x^{2} \left(a + 2\right) - x 2 \left(a + 3\right)\right)\right) + 5 = 0$$
Obtenemos la ecuación cuadrática
$$a x^{2} - 2 a x + a + 2 x^{2} - 6 x + 5 = 0$$
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
False
$$b = - 2 a - 6$$
$$c = a + 5$$
, entonces
D = b^2 - 4 * a * c =
(-6 - 2*a)^2 - 4 * (2 + a) * (5 + a) = (-6 - 2*a)^2 - (5 + a)*(8 + 4*a)
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{2 a + \sqrt{\left(- 2 a - 6\right)^{2} - \left(a + 5\right) \left(4 a + 8\right)} + 6}{2 a + 4}$$
$$x_{2} = \frac{2 a - \sqrt{\left(- 2 a - 6\right)^{2} - \left(a + 5\right) \left(4 a + 8\right)} + 6}{2 a + 4}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$a + x^{2} \left(a + 2\right) - x \left(2 a + 6\right) + 5 = 0$$
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
$$2 - x^{2} = 0$$
su solución
$$x = - \sqrt{2}$$
$$x = \sqrt{2}$$
Con
$$a = -2$$
la ecuación será
$$3 - 2 x = 0$$
su solución
$$x = \frac{3}{2}$$
$$a + x^{2} \left(a + 2\right) - x \left(2 a + 6\right) + 5 = 0$$
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
$$2 - x^{2} = 0$$
su solución
$$x = - \sqrt{2}$$
$$x = \sqrt{2}$$
Con
$$a = -2$$
la ecuación será
$$3 - 2 x = 0$$
su solución
$$x = \frac{3}{2}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(a + \left(x^{2} \left(a + 2\right) - x 2 \left(a + 3\right)\right)\right) + 5 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a + x^{2} \left(a + 2\right) - x \left(2 a + 6\right) + 5}{a + 2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{- 2 a - 6}{a + 2}$$
$$q = \frac{c}{a}$$
$$q = \frac{a + 5}{a + 2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{- 2 a - 6}{a + 2}$$
$$x_{1} x_{2} = \frac{a + 5}{a + 2}$$
$$\left(a + \left(x^{2} \left(a + 2\right) - x 2 \left(a + 3\right)\right)\right) + 5 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a + x^{2} \left(a + 2\right) - x \left(2 a + 6\right) + 5}{a + 2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{- 2 a - 6}{a + 2}$$
$$q = \frac{c}{a}$$
$$q = \frac{a + 5}{a + 2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{- 2 a - 6}{a + 2}$$
$$x_{1} x_{2} = \frac{a + 5}{a + 2}$$
Gráfica
Respuesta rápida
[src]
/ / ________________________ \ / ________________________ \ \ / ________________________ \ / ________________________ \
| | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ |
|(2 + re(a))*|- \/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)| |3 - \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)|*im(a)| (2 + re(a))*|3 - \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)| |- \/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)|*im(a)
| \ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / /
x1 = I*|---------------------------------------------------------------------------------- - ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ + ----------------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ (2 + re(a)) + im (a) (2 + re(a)) + im (a) / (2 + re(a)) + im (a) (2 + re(a)) + im (a)
$$x_{1} = \frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
/ / ________________________ \ / ________________________ \ \ / ________________________ \ / ________________________ \
| |4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | |4 / 2 2 /atan2(-im(a), -1 - re(a))\ |
|(2 + re(a))*|\/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)| |3 + \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)|*im(a)| (2 + re(a))*|3 + \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)| |\/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)|*im(a)
| \ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / /
x2 = I*|-------------------------------------------------------------------------------- - ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ + --------------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ (2 + re(a)) + im (a) (2 + re(a)) + im (a) / (2 + re(a)) + im (a) (2 + re(a)) + im (a)
$$x_{2} = \frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
x2 = (((-re(a) - 1)^2 + im(a)^2)^(1/4)*sin(atan2(-im(a, -re(a) - 1)/2) + im(a))*im(a)/((re(a) + 2)^2 + im(a)^2) + i*((((-re(a) - 1)^2 + im(a)^2)^(1/4)*sin(atan2(-im(a), -re(a) - 1)/2) + im(a))*(re(a) + 2)/((re(a) + 2)^2 + im(a)^2) - (((-re(a) - 1)^2 + im(a)^2)^(1/4)*cos(atan2(-im(a), -re(a) - 1)/2) + re(a) + 3)*im(a)/((re(a) + 2)^2 + im(a)^2)) + (re(a) + 2)*(((-re(a) - 1)^2 + im(a)^2)^(1/4)*cos(atan2(-im(a), -re(a) - 1)/2) + re(a) + 3)/((re(a) + 2)^2 + im(a)^2))
Suma y producto de raíces
[src]
suma
/ / ________________________ \ / ________________________ \ \ / ________________________ \ / ________________________ \ / / ________________________ \ / ________________________ \ \ / ________________________ \ / ________________________ \ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | |4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | |4 / 2 2 /atan2(-im(a), -1 - re(a))\ | |(2 + re(a))*|- \/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)| |3 - \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)|*im(a)| (2 + re(a))*|3 - \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)| |- \/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)|*im(a) |(2 + re(a))*|\/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)| |3 + \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)|*im(a)| (2 + re(a))*|3 + \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)| |\/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)|*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 + re(a)) + im (a) (2 + re(a)) + im (a) / (2 + re(a)) + im (a) (2 + re(a)) + im (a) \ (2 + re(a)) + im (a) (2 + re(a)) + im (a) / (2 + re(a)) + im (a) (2 + re(a)) + im (a)
$$\left(\frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right) \operatorname{im}{\left(a\right)}}{\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(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \left(\frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right) \operatorname{im}{\left(a\right)}}{\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(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)$$
=
/ / ________________________ \ / ________________________ \ \ / / ________________________ \ / ________________________ \ \ / ________________________ \ / ________________________ \ / ________________________ \ / ________________________ \ | |4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | |4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | |(2 + re(a))*|\/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)| |3 + \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)|*im(a)| |(2 + re(a))*|- \/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)| |3 - \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)|*im(a)| (2 + re(a))*|3 + \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)| (2 + re(a))*|3 - \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)| |\/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)|*im(a) |- \/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)|*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 + re(a)) + im (a) (2 + re(a)) + im (a) / \ (2 + re(a)) + im (a) (2 + re(a)) + im (a) / (2 + re(a)) + im (a) (2 + re(a)) + im (a) (2 + re(a)) + im (a) (2 + re(a)) + im (a)
$$\frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right) \operatorname{im}{\left(a\right)}}{\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(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right) \operatorname{im}{\left(a\right)}}{\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(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{re}{\left(a\right)} + 2\right) \left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
producto
/ / / ________________________ \ / ________________________ \ \ / ________________________ \ / ________________________ \ \ / / / ________________________ \ / ________________________ \ \ / ________________________ \ / ________________________ \ \ | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | | |4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | 4 / 2 2 /atan2(-im(a), -1 - re(a))\ | |4 / 2 2 /atan2(-im(a), -1 - re(a))\ | | | |(2 + re(a))*|- \/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)| |3 - \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)|*im(a)| (2 + re(a))*|3 - \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)| |- \/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)|*im(a)| | |(2 + re(a))*|\/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)| |3 + \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)|*im(a)| (2 + re(a))*|3 + \/ (-1 - re(a)) + im (a) *cos|-------------------------| + re(a)| |\/ (-1 - re(a)) + im (a) *sin|-------------------------| + im(a)|*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 + re(a)) + im (a) (2 + re(a)) + im (a) / (2 + re(a)) + im (a) (2 + re(a)) + im (a) / \ \ (2 + re(a)) + im (a) (2 + re(a)) + im (a) / (2 + re(a)) + im (a) (2 + re(a)) + im (a) /
$$\left(\frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(- \sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right) \operatorname{im}{\left(a\right)}}{\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(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) \left(\frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 2\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\sqrt[4]{\left(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right) \operatorname{im}{\left(a\right)}}{\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(- \operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} - 1 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)} + 3\right)}{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)$$
=
2 2
10 + im (a) + re (a) + 7*re(a) - 3*I*im(a)
------------------------------------------
2 2
4 + im (a) + re (a) + 4*re(a)
$$\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 3 i \operatorname{im}{\left(a\right)} + 10}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 4}$$
(10 + im(a)^2 + re(a)^2 + 7*re(a) - 3*i*im(a))/(4 + im(a)^2 + re(a)^2 + 4*re(a))