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- La ecuación:
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Ecuación (x+9)^2=(x+6)^2
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Ecuación 9,4x-7,8x+0,52=1
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Ecuación 2x^2+5x-7=0
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Ecuación x^2-8*x+12=0
- Expresar {x} en función de y en la ecuación:
- 17*x-18*y=10
- 17*x+16*y=7
- 15*x-3*y=-8
- 5*x+2*y=-14
- Forma canónica:
- xy+xz+yz
- Expresión:
- xy+xz+yz
-
Expresiones idénticas
- xy+xz+yz= treinta -x^ dos
- xy más xz más yz es igual a 30 menos x al cuadrado
- xy más xz más yz es igual a treinta menos x en el grado dos
- xy+xz+yz=30-x2
- xy+xz+yz=30-x²
- xy+xz+yz=30-x en el grado 2
-
Expresiones semejantes
- xy+xz+yz=30+x^2
- xy+xz-yz=30-x^2
- xy-xz+yz=30-x^2
-
Expresiones con funciones
- xy
- xy-y=0
- xy-3y=-1
- xy+48y=10,2
- xy+2y=0
- xysin(x)x
- xz
- xz
- xz+x=36
xy+xz+yz=30-x^2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$y z + \left(x y + x z\right) = 30 - x^{2}$$
en
$$\left(x^{2} - 30\right) + \left(y z + \left(x y + x z\right)\right) = 0$$
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
$$a = 1$$
$$b = y + z$$
$$c = y z - 30$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = - \frac{y}{2} - \frac{z}{2} + \frac{\sqrt{- 4 y z + \left(y + z\right)^{2} + 120}}{2}$$
$$x_{2} = - \frac{y}{2} - \frac{z}{2} - \frac{\sqrt{- 4 y z + \left(y + z\right)^{2} + 120}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$y z + \left(x y + x z\right) = 30 - x^{2}$$
en
$$\left(x^{2} - 30\right) + \left(y z + \left(x y + x z\right)\right) = 0$$
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
$$a = 1$$
$$b = y + z$$
$$c = y z - 30$$
, entonces
D = b^2 - 4 * a * c =
(y + z)^2 - 4 * (1) * (-30 + y*z) = 120 + (y + z)^2 - 4*y*z
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = - \frac{y}{2} - \frac{z}{2} + \frac{\sqrt{- 4 y z + \left(y + z\right)^{2} + 120}}{2}$$
$$x_{2} = - \frac{y}{2} - \frac{z}{2} - \frac{\sqrt{- 4 y z + \left(y + z\right)^{2} + 120}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = y + z$$
$$q = \frac{c}{a}$$
$$q = y z - 30$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - y - z$$
$$x_{1} x_{2} = y z - 30$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = y + z$$
$$q = \frac{c}{a}$$
$$q = y z - 30$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - y - z$$
$$x_{1} x_{2} = y z - 30$$
Gráfica
Respuesta rápida
[src]
/ ________________________________________________________________________________________________________ \ ________________________________________________________________________________________________________
| / 2 / / 2 2 2 2 \\| / 2 / / 2 2 2 2 \\
| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|
| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *sin|------------------------------------------------------------------------------------------------------|| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *cos|------------------------------------------------------------------------------------------------------|
re(y) re(z) | im(y) im(z) \ 2 /| \ 2 /
x1 = - ----- - ----- + I*|- ----- - ----- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 2 / 2
$$x_{1} = - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} + i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2} - \frac{\operatorname{im}{\left(z\right)}}{2}\right) - \frac{\operatorname{re}{\left(y\right)}}{2} - \frac{\operatorname{re}{\left(z\right)}}{2}$$
/ ________________________________________________________________________________________________________ \ ________________________________________________________________________________________________________
| / 2 / / 2 2 2 2 \\| / 2 / / 2 2 2 2 \\
| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|
| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *sin|------------------------------------------------------------------------------------------------------|| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *cos|------------------------------------------------------------------------------------------------------|
re(y) re(z) | im(y) im(z) \ 2 /| \ 2 /
x2 = - ----- - ----- + I*|- ----- - ----- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 2 / 2
$$x_{2} = \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} + i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2} - \frac{\operatorname{im}{\left(z\right)}}{2}\right) - \frac{\operatorname{re}{\left(y\right)}}{2} - \frac{\operatorname{re}{\left(z\right)}}{2}$$
x2 = ((2*re(y)*im(y) + 2*re(z)*im(z) - 2*im(y*z))^2 + (re(y)^2 + re(z)^2 - 2*re(y*z) - im(y)^2 - im(z)^2 + 120)^2)^(1/4)*cos(atan2(2*re(y)*im(y) + 2*re(z)*im(z) - 2*im(y*z, re(y)^2 + re(z)^2 - 2*re(y*z) - im(y)^2 - im(z)^2 + 120)/2)/2 + i*(((2*re(y)*im(y) + 2*re(z)*im(z) - 2*im(y*z))^2 + (re(y)^2 + re(z)^2 - 2*re(y*z) - im(y)^2 - im(z)^2 + 120)^2)^(1/4)*sin(atan2(2*re(y)*im(y) + 2*re(z)*im(z) - 2*im(y*z), re(y)^2 + re(z)^2 - 2*re(y*z) - im(y)^2 - im(z)^2 + 120)/2)/2 - im(y)/2 - im(z)/2) - re(y)/2 - re(z)/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 \\
| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/| | 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|
| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *sin|------------------------------------------------------------------------------------------------------|| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *cos|------------------------------------------------------------------------------------------------------| | \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *sin|------------------------------------------------------------------------------------------------------|| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *cos|------------------------------------------------------------------------------------------------------|
re(y) re(z) | im(y) im(z) \ 2 /| \ 2 / re(y) re(z) | im(y) im(z) \ 2 /| \ 2 /
- ----- - ----- + I*|- ----- - ----- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + - ----- - ----- + I*|- ----- - ----- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 2 / 2 2 2 \ 2 2 2 / 2
$$\left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} + i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2} - \frac{\operatorname{im}{\left(z\right)}}{2}\right) - \frac{\operatorname{re}{\left(y\right)}}{2} - \frac{\operatorname{re}{\left(z\right)}}{2}\right) + \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} + i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2} - \frac{\operatorname{im}{\left(z\right)}}{2}\right) - \frac{\operatorname{re}{\left(y\right)}}{2} - \frac{\operatorname{re}{\left(z\right)}}{2}\right)$$
=
/ ________________________________________________________________________________________________________ \ / ________________________________________________________________________________________________________ \
| / 2 / / 2 2 2 2 \\| | / 2 / / 2 2 2 2 \\|
| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| | 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/||
| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *sin|------------------------------------------------------------------------------------------------------|| | \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *sin|------------------------------------------------------------------------------------------------------||
| im(y) im(z) \ 2 /| | im(y) im(z) \ 2 /|
-re(y) - re(z) + I*|- ----- - ----- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + I*|- ----- - ----- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\ 2 2 2 / \ 2 2 2 /
$$i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2} - \frac{\operatorname{im}{\left(z\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2} - \frac{\operatorname{im}{\left(z\right)}}{2}\right) - \operatorname{re}{\left(y\right)} - \operatorname{re}{\left(z\right)}$$
producto
/ / ________________________________________________________________________________________________________ \ ________________________________________________________________________________________________________ \ / / ________________________________________________________________________________________________________ \ ________________________________________________________________________________________________________ \ | | / 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 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| | | 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| 4 / 2 / 2 2 2 2 \ |atan2\-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z), 120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/|| | | \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *sin|------------------------------------------------------------------------------------------------------|| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *cos|------------------------------------------------------------------------------------------------------|| | | \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *sin|------------------------------------------------------------------------------------------------------|| \/ (-2*im(y*z) + 2*im(y)*re(y) + 2*im(z)*re(z)) + \120 + re (y) + re (z) - im (y) - im (z) - 2*re(y*z)/ *cos|------------------------------------------------------------------------------------------------------|| | re(y) re(z) | im(y) im(z) \ 2 /| \ 2 /| | re(y) re(z) | im(y) im(z) \ 2 /| \ 2 /| |- ----- - ----- + I*|- ----- - ----- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|- ----- - ----- + I*|- ----- - ----- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| \ 2 2 \ 2 2 2 / 2 / \ 2 2 \ 2 2 2 / 2 /
$$\left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} + i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2} - \frac{\operatorname{im}{\left(z\right)}}{2}\right) - \frac{\operatorname{re}{\left(y\right)}}{2} - \frac{\operatorname{re}{\left(z\right)}}{2}\right) \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} + i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(z\right)} \operatorname{im}{\left(z\right)} - 2 \operatorname{im}{\left(y z\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{re}{\left(z\right)}\right)^{2} - 2 \operatorname{re}{\left(y z\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(z\right)}\right)^{2} + 120 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2} - \frac{\operatorname{im}{\left(z\right)}}{2}\right) - \frac{\operatorname{re}{\left(y\right)}}{2} - \frac{\operatorname{re}{\left(z\right)}}{2}\right)$$
=
re(y*z) I*im(y*z) re(y)*re(z) im(y)*im(z) I*im(y)*re(z) I*im(z)*re(y)
-30 + ------- + --------- + ----------- - ----------- + ------------- + -------------
2 2 2 2 2 2
$$\frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(z\right)}}{2} + \frac{i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(z\right)}}{2} + \frac{i \operatorname{re}{\left(z\right)} \operatorname{im}{\left(y\right)}}{2} + \frac{\operatorname{re}{\left(y z\right)}}{2} - \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(z\right)}}{2} + \frac{i \operatorname{im}{\left(y z\right)}}{2} - 30$$
-30 + re(y*z)/2 + i*im(y*z)/2 + re(y)*re(z)/2 - im(y)*im(z)/2 + i*im(y)*re(z)/2 + i*im(z)*re(y)/2