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- ax2-bx+с=0
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ax2+bx+с=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{- b + \sqrt{- 4 a c + b^{2}}}{2 a}$$
$$x_{2} = \frac{- b - \sqrt{- 4 a c + b^{2}}}{2 a}$$
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
True
True
True
, entonces
D = b^2 - 4 * a * c =
(b)^2 - 4 * (a) * (c) = b^2 - 4*a*c
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{- b + \sqrt{- 4 a c + b^{2}}}{2 a}$$
$$x_{2} = \frac{- b - \sqrt{- 4 a c + b^{2}}}{2 a}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$a x^{2} + b x + c = 0$$
Коэффициент при x равен
$$a$$
entonces son posibles los casos para a :
$$a < 0$$
$$a = 0$$
Consideremos todos los casos con detalles:
Con
$$a < 0$$
la ecuación será
$$b x + c - x^{2} = 0$$
su solución
$$x = \frac{b}{2} - \frac{\sqrt{b^{2} + 4 c}}{2}$$
$$x = \frac{b}{2} + \frac{\sqrt{b^{2} + 4 c}}{2}$$
Con
$$a = 0$$
la ecuación será
$$b x + c = 0$$
su solución
$$x = - \frac{c}{b}$$
$$a x^{2} + b x + c = 0$$
Коэффициент при x равен
$$a$$
entonces son posibles los casos para a :
$$a < 0$$
$$a = 0$$
Consideremos todos los casos con detalles:
Con
$$a < 0$$
la ecuación será
$$b x + c - x^{2} = 0$$
su solución
$$x = \frac{b}{2} - \frac{\sqrt{b^{2} + 4 c}}{2}$$
$$x = \frac{b}{2} + \frac{\sqrt{b^{2} + 4 c}}{2}$$
Con
$$a = 0$$
la ecuación será
$$b x + c = 0$$
su solución
$$x = - \frac{c}{b}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$c + \left(a x^{2} + b x\right) = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + b x + c}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{b}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{b}{a}$$
$$x_{1} x_{2} = \frac{c}{a}$$
$$c + \left(a x^{2} + b x\right) = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + b x + c}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{b}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{b}{a}$$
$$x_{1} x_{2} = \frac{c}{a}$$
Gráfica
Respuesta rápida
[src]
// ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \
|| / 2 / / 2 2 \\| | / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
|| 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/||
||-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*re(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*im(a)| |-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*im(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*re(a)
|\ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 //
x1 = I*|--------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------| + --------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$x_{1} = \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
// ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \
|| / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | | | / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ |
||4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| |
||\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*im(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*re(a)| |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*re(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*im(a)
|\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / /
x2 = I*|-------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$x_{2} = - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
x2 = -(((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*sin(atan2(2*re(b)*im(b) - 4*im(a*c, re(b)^2 - 4*re(a*c) - im(b)^2)/2) + im(b))*im(a)/(2*(re(a)^2 + im(a)^2)) - (((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*cos(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + re(b))*re(a)/(2*(re(a)^2 + im(a)^2)) + i*(-(((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*sin(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + im(b))*re(a)/(2*(re(a)^2 + im(a)^2)) + (((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*cos(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + re(b))*im(a)/(2*(re(a)^2 + im(a)^2))))
Suma y producto de raíces
[src]
suma
// ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \ // ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \ || / 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 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| ||4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | ||-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*re(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*im(a)| |-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*im(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*re(a) ||\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*im(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*re(a)| |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*re(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*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*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$\left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)\right) + \left(- \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)\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 \\ | || 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | ||4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | | | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | ||-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*re(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*im(a)| ||\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*im(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*re(a)| |-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*im(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*re(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*re(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*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*\im (a) + re (a)/ 2*\im (a) + re (a)/ / \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + i \left(- \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
producto
/ // ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \ \ / // ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \ \ | || / 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 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | | ||4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | | | ||-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*re(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*im(a)| |-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*im(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*re(a)| | ||\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*im(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*re(a)| |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*re(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*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*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / \ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ /
$$\left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)\right) \left(- \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)\right)$$
=
2 2 2 2
re (a)*re(a*c) - im (a)*re(a*c) + I*re (a)*im(a*c) - I*im (a)*im(a*c) + 2*im(a)*im(a*c)*re(a) - 2*I*im(a)*re(a)*re(a*c)
-----------------------------------------------------------------------------------------------------------------------
4 4 2 2
im (a) + re (a) + 2*im (a)*re (a)
$$\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{re}{\left(a c\right)} + i \left(\operatorname{re}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a c\right)} - 2 i \operatorname{re}{\left(a\right)} \operatorname{re}{\left(a c\right)} \operatorname{im}{\left(a\right)} + 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} \operatorname{im}{\left(a c\right)} - \operatorname{re}{\left(a c\right)} \left(\operatorname{im}{\left(a\right)}\right)^{2} - i \left(\operatorname{im}{\left(a\right)}\right)^{2} \operatorname{im}{\left(a c\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{4} + 2 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{4}}$$
(re(a)^2*re(a*c) - im(a)^2*re(a*c) + i*re(a)^2*im(a*c) - i*im(a)^2*im(a*c) + 2*im(a)*im(a*c)*re(a) - 2*i*im(a)*re(a)*re(a*c))/(im(a)^4 + re(a)^4 + 2*im(a)^2*re(a)^2)