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-11x1^2-6x2^2-6x3^2+12x1x2-102x1x3+6x2x3 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 2 - 11*x1 - 6*x2 - 6*x3 + 12*x1*x2 - 102*x1*x3 + 6*x2*x3 = 0
$$6 x_{2} x_{3} + \left(- 102 x_{1} x_{3} + \left(12 x_{1} x_{2} + \left(- 6 x_{3}^{2} + \left(- 11 x_{1}^{2} - 6 x_{2}^{2}\right)\right)\right)\right) = 0$$
Solución detallada
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{31} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{32} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -6$$
$$b = - 102 x_{1} + 6 x_{2}$$
$$c = - 11 x_{1}^{2} + 12 x_{1} x_{2} - 6 x_{2}^{2}$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{31} = - \frac{17 x_{1}}{2} + \frac{x_{2}}{2} - \frac{\sqrt{- 264 x_{1}^{2} + 288 x_{1} x_{2} - 144 x_{2}^{2} + \left(- 102 x_{1} + 6 x_{2}\right)^{2}}}{12}$$
$$x_{32} = - \frac{17 x_{1}}{2} + \frac{x_{2}}{2} + \frac{\sqrt{- 264 x_{1}^{2} + 288 x_{1} x_{2} - 144 x_{2}^{2} + \left(- 102 x_{1} + 6 x_{2}\right)^{2}}}{12}$$
a*x3^2 + b*x3 + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{31} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{32} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -6$$
$$b = - 102 x_{1} + 6 x_{2}$$
$$c = - 11 x_{1}^{2} + 12 x_{1} x_{2} - 6 x_{2}^{2}$$
, entonces
D = b^2 - 4 * a * c =
(-102*x1 + 6*x2)^2 - 4 * (-6) * (-11*x1^2 - 6*x2^2 + 12*x1*x2) = (-102*x1 + 6*x2)^2 - 264*x1^2 - 144*x2^2 + 288*x1*x2
La ecuación tiene dos raíces.
x31 = (-b + sqrt(D)) / (2*a)
x32 = (-b - sqrt(D)) / (2*a)
o
$$x_{31} = - \frac{17 x_{1}}{2} + \frac{x_{2}}{2} - \frac{\sqrt{- 264 x_{1}^{2} + 288 x_{1} x_{2} - 144 x_{2}^{2} + \left(- 102 x_{1} + 6 x_{2}\right)^{2}}}{12}$$
$$x_{32} = - \frac{17 x_{1}}{2} + \frac{x_{2}}{2} + \frac{\sqrt{- 264 x_{1}^{2} + 288 x_{1} x_{2} - 144 x_{2}^{2} + \left(- 102 x_{1} + 6 x_{2}\right)^{2}}}{12}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$6 x_{2} x_{3} + \left(- 102 x_{1} x_{3} + \left(12 x_{1} x_{2} + \left(- 6 x_{3}^{2} + \left(- 11 x_{1}^{2} - 6 x_{2}^{2}\right)\right)\right)\right) = 0$$
de
$$a x_{3}^{2} + b x_{3} + c = 0$$
como ecuación cuadrática reducida
$$x_{3}^{2} + \frac{b x_{3}}{a} + \frac{c}{a} = 0$$
$$\frac{11 x_{1}^{2}}{6} - 2 x_{1} x_{2} + 17 x_{1} x_{3} + x_{2}^{2} - x_{2} x_{3} + x_{3}^{2} = 0$$
$$p x_{3} + q + x_{3}^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 17 x_{1} - x_{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{11 x_{1}^{2}}{6} - 2 x_{1} x_{2} + x_{2}^{2}$$
Fórmulas de Cardano-Vieta
$$x_{3 1} + x_{3 2} = - p$$
$$x_{3 1} x_{3 2} = q$$
$$x_{3 1} + x_{3 2} = - 17 x_{1} + x_{2}$$
$$x_{3 1} x_{3 2} = \frac{11 x_{1}^{2}}{6} - 2 x_{1} x_{2} + x_{2}^{2}$$
$$6 x_{2} x_{3} + \left(- 102 x_{1} x_{3} + \left(12 x_{1} x_{2} + \left(- 6 x_{3}^{2} + \left(- 11 x_{1}^{2} - 6 x_{2}^{2}\right)\right)\right)\right) = 0$$
de
$$a x_{3}^{2} + b x_{3} + c = 0$$
como ecuación cuadrática reducida
$$x_{3}^{2} + \frac{b x_{3}}{a} + \frac{c}{a} = 0$$
$$\frac{11 x_{1}^{2}}{6} - 2 x_{1} x_{2} + 17 x_{1} x_{3} + x_{2}^{2} - x_{2} x_{3} + x_{3}^{2} = 0$$
$$p x_{3} + q + x_{3}^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 17 x_{1} - x_{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{11 x_{1}^{2}}{6} - 2 x_{1} x_{2} + x_{2}^{2}$$
Fórmulas de Cardano-Vieta
$$x_{3 1} + x_{3 2} = - p$$
$$x_{3 1} x_{3 2} = q$$
$$x_{3 1} + x_{3 2} = - 17 x_{1} + x_{2}$$
$$x_{3 1} x_{3 2} = \frac{11 x_{1}^{2}}{6} - 2 x_{1} x_{2} + x_{2}^{2}$$
Gráfica
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\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/| | 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|
| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------|| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------| | \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------|| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------|
re(x2) 17*re(x1) |im(x2) 17*im(x1) \ 2 /| \ 2 / re(x2) 17*re(x1) |im(x2) 17*im(x1) \ 2 /| \ 2 /
------ - --------- + I*|------ - --------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------ - --------- + I*|------ - --------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 6 / 6 2 2 \ 2 2 6 / 6
$$\left(- \frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} + i \left(- \frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} - \frac{17 \operatorname{im}{\left(x_{1}\right)}}{2} + \frac{\operatorname{im}{\left(x_{2}\right)}}{2}\right) - \frac{17 \operatorname{re}{\left(x_{1}\right)}}{2} + \frac{\operatorname{re}{\left(x_{2}\right)}}{2}\right) + \left(\frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} + i \left(\frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} - \frac{17 \operatorname{im}{\left(x_{1}\right)}}{2} + \frac{\operatorname{im}{\left(x_{2}\right)}}{2}\right) - \frac{17 \operatorname{re}{\left(x_{1}\right)}}{2} + \frac{\operatorname{re}{\left(x_{2}\right)}}{2}\right)$$
=
/ ________________________________________________________________________________________________________________________________________ \ / ________________________________________________________________________________________________________________________________________ \
| / 2 / / 2 2 2 2 \\| | / 2 / / 2 2 2 2 \\|
| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| | 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/||
| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------|| | \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------||
|im(x2) 17*im(x1) \ 2 /| |im(x2) 17*im(x1) \ 2 /|
-17*re(x1) + I*|------ - --------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + I*|------ - --------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + re(x2)
\ 2 2 6 / \ 2 2 6 /
$$i \left(- \frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} - \frac{17 \operatorname{im}{\left(x_{1}\right)}}{2} + \frac{\operatorname{im}{\left(x_{2}\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} - \frac{17 \operatorname{im}{\left(x_{1}\right)}}{2} + \frac{\operatorname{im}{\left(x_{2}\right)}}{2}\right) - 17 \operatorname{re}{\left(x_{1}\right)} + \operatorname{re}{\left(x_{2}\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\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| | | 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| | | \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------|| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------|| | | \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------|| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------|| |re(x2) 17*re(x1) |im(x2) 17*im(x1) \ 2 /| \ 2 /| |re(x2) 17*re(x1) |im(x2) 17*im(x1) \ 2 /| \ 2 /| |------ - --------- + I*|------ - --------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|------ - --------- + I*|------ - --------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| \ 2 2 \ 2 2 6 / 6 / \ 2 2 \ 2 2 6 / 6 /
$$\left(- \frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} + i \left(- \frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} - \frac{17 \operatorname{im}{\left(x_{1}\right)}}{2} + \frac{\operatorname{im}{\left(x_{2}\right)}}{2}\right) - \frac{17 \operatorname{re}{\left(x_{1}\right)}}{2} + \frac{\operatorname{re}{\left(x_{2}\right)}}{2}\right) \left(\frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} + i \left(\frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} - \frac{17 \operatorname{im}{\left(x_{1}\right)}}{2} + \frac{\operatorname{im}{\left(x_{2}\right)}}{2}\right) - \frac{17 \operatorname{re}{\left(x_{1}\right)}}{2} + \frac{\operatorname{re}{\left(x_{2}\right)}}{2}\right)$$
=
2 2
2 2 11*im (x1) 11*re (x1) 13*re(x1*x2) 17*re(x1)*re(x2) 13*I*im(x1*x2) 17*im(x1)*im(x2) 17*I*im(x1)*re(x2) 17*I*im(x2)*re(x1) 11*I*im(x1)*re(x1)
re (x2) - im (x2) - ---------- + ---------- + ------------ - ---------------- + -------------- + ---------------- + 2*I*im(x2)*re(x2) - ------------------ - ------------------ + ------------------
6 6 2 2 2 2 2 2 3
$$\frac{11 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2}}{6} - \frac{17 \operatorname{re}{\left(x_{1}\right)} \operatorname{re}{\left(x_{2}\right)}}{2} + \frac{11 i \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)}}{3} - \frac{17 i \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{2}\right)}}{2} + \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - \frac{17 i \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{1}\right)}}{2} + 2 i \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} + \frac{13 \operatorname{re}{\left(x_{1} x_{2}\right)}}{2} - \frac{11 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2}}{6} + \frac{17 \operatorname{im}{\left(x_{1}\right)} \operatorname{im}{\left(x_{2}\right)}}{2} - \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} + \frac{13 i \operatorname{im}{\left(x_{1} x_{2}\right)}}{2}$$
re(x2)^2 - im(x2)^2 - 11*im(x1)^2/6 + 11*re(x1)^2/6 + 13*re(x1*x2)/2 - 17*re(x1)*re(x2)/2 + 13*i*im(x1*x2)/2 + 17*im(x1)*im(x2)/2 + 2*i*im(x2)*re(x2) - 17*i*im(x1)*re(x2)/2 - 17*i*im(x2)*re(x1)/2 + 11*i*im(x1)*re(x1)/3
Respuesta rápida
[src]
/ ________________________________________________________________________________________________________________________________________ \ ________________________________________________________________________________________________________________________________________
| / 2 / / 2 2 2 2 \\| / 2 / / 2 2 2 2 \\
| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|
| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------|| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------|
re(x2) 17*re(x1) |im(x2) 17*im(x1) \ 2 /| \ 2 /
x31 = ------ - --------- + I*|------ - --------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 6 / 6
$$x_{31} = - \frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} + i \left(- \frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} - \frac{17 \operatorname{im}{\left(x_{1}\right)}}{2} + \frac{\operatorname{im}{\left(x_{2}\right)}}{2}\right) - \frac{17 \operatorname{re}{\left(x_{1}\right)}}{2} + \frac{\operatorname{re}{\left(x_{2}\right)}}{2}$$
/ ________________________________________________________________________________________________________________________________________ \ ________________________________________________________________________________________________________________________________________
| / 2 / / 2 2 2 2 \\| / 2 / / 2 2 2 2 \\
| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|| 4 / 2 / 2 2 2 2 \ |atan2\-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1), - 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/|
| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *sin|--------------------------------------------------------------------------------------------------------------------------------------|| \/ (-234*im(x1*x2) - 54*im(x2)*re(x2) + 5070*im(x1)*re(x1)) + \- 2535*im (x1) - 234*re(x1*x2) - 27*re (x2) + 27*im (x2) + 2535*re (x1)/ *cos|--------------------------------------------------------------------------------------------------------------------------------------|
re(x2) 17*re(x1) |im(x2) 17*im(x1) \ 2 /| \ 2 /
x32 = ------ - --------- + I*|------ - --------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 6 / 6
$$x_{32} = \frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} + i \left(\frac{\sqrt[4]{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)}\right)^{2} + \left(2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(5070 \operatorname{re}{\left(x_{1}\right)} \operatorname{im}{\left(x_{1}\right)} - 54 \operatorname{re}{\left(x_{2}\right)} \operatorname{im}{\left(x_{2}\right)} - 234 \operatorname{im}{\left(x_{1} x_{2}\right)},2535 \left(\operatorname{re}{\left(x_{1}\right)}\right)^{2} - 27 \left(\operatorname{re}{\left(x_{2}\right)}\right)^{2} - 234 \operatorname{re}{\left(x_{1} x_{2}\right)} - 2535 \left(\operatorname{im}{\left(x_{1}\right)}\right)^{2} + 27 \left(\operatorname{im}{\left(x_{2}\right)}\right)^{2} \right)}}{2} \right)}}{6} - \frac{17 \operatorname{im}{\left(x_{1}\right)}}{2} + \frac{\operatorname{im}{\left(x_{2}\right)}}{2}\right) - \frac{17 \operatorname{re}{\left(x_{1}\right)}}{2} + \frac{\operatorname{re}{\left(x_{2}\right)}}{2}$$
x32 = ((5070*re(x1)*im(x1) - 54*re(x2)*im(x2) - 234*im(x1*x2))^2 + (2535*re(x1)^2 - 27*re(x2)^2 - 234*re(x1*x2) - 2535*im(x1)^2 + 27*im(x2)^2)^2)^(1/4)*cos(atan2(5070*re(x1)*im(x1) - 54*re(x2)*im(x2) - 234*im(x1*x2, 2535*re(x1)^2 - 27*re(x2)^2 - 234*re(x1*x2) - 2535*im(x1)^2 + 27*im(x2)^2)/2)/6 + i*(((5070*re(x1)*im(x1) - 54*re(x2)*im(x2) - 234*im(x1*x2))^2 + (2535*re(x1)^2 - 27*re(x2)^2 - 234*re(x1*x2) - 2535*im(x1)^2 + 27*im(x2)^2)^2)^(1/4)*sin(atan2(5070*re(x1)*im(x1) - 54*re(x2)*im(x2) - 234*im(x1*x2), 2535*re(x1)^2 - 27*re(x2)^2 - 234*re(x1*x2) - 2535*im(x1)^2 + 27*im(x2)^2)/2)/6 - 17*im(x1)/2 + im(x2)/2) - 17*re(x1)/2 + re(x2)/2)