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Expresiones idénticas
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Expresiones semejantes
- sqrt(x-2)+sqrt(x+6)=2
- t(x)=2/3x-2/x+3/2
- sqrt(x-2)+sqrt(4-x)=x^2-6*x+11
- t(x)=2/3x+2/x-3/2
- x^2-5=sqrt(x+5)
t(x)=2/3x-2/x-3/2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2*x 2 3
t*x = --- - - - -
3 x 2
$$t x = \left(\frac{2 x}{3} - \frac{2}{x}\right) - \frac{3}{2}$$
Solución detallada
Tenemos la ecuación:
$$t x = \left(\frac{2 x}{3} - \frac{2}{x}\right) - \frac{3}{2}$$
Multipliquemos las dos partes de la ecuación por los denominadores:
y x
obtendremos:
$$x t x = x \left(\left(\frac{2 x}{3} - \frac{2}{x}\right) - \frac{3}{2}\right)$$
$$t x^{2} = \frac{2 x^{2}}{3} - \frac{3 x}{2} - 2$$
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$t x^{2} = \frac{2 x^{2}}{3} - \frac{3 x}{2} - 2$$
en
$$t x^{2} - \frac{2 x^{2}}{3} + \frac{3 x}{2} + 2 = 0$$
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
$$a = t - \frac{2}{3}$$
$$b = \frac{3}{2}$$
$$c = 2$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{\frac{91}{12} - 8 t} - \frac{3}{2}}{2 t - \frac{4}{3}}$$
$$x_{2} = \frac{- \sqrt{\frac{91}{12} - 8 t} - \frac{3}{2}}{2 t - \frac{4}{3}}$$
$$t x = \left(\frac{2 x}{3} - \frac{2}{x}\right) - \frac{3}{2}$$
Multipliquemos las dos partes de la ecuación por los denominadores:
y x
obtendremos:
$$x t x = x \left(\left(\frac{2 x}{3} - \frac{2}{x}\right) - \frac{3}{2}\right)$$
$$t x^{2} = \frac{2 x^{2}}{3} - \frac{3 x}{2} - 2$$
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$t x^{2} = \frac{2 x^{2}}{3} - \frac{3 x}{2} - 2$$
en
$$t x^{2} - \frac{2 x^{2}}{3} + \frac{3 x}{2} + 2 = 0$$
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
$$a = t - \frac{2}{3}$$
$$b = \frac{3}{2}$$
$$c = 2$$
, entonces
D = b^2 - 4 * a * c =
(3/2)^2 - 4 * (-2/3 + t) * (2) = 91/12 - 8*t
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{\frac{91}{12} - 8 t} - \frac{3}{2}}{2 t - \frac{4}{3}}$$
$$x_{2} = \frac{- \sqrt{\frac{91}{12} - 8 t} - \frac{3}{2}}{2 t - \frac{4}{3}}$$
Gráfica
Respuesta rápida
[src]
/ / ___________________________________ \ \ / ___________________________________ \
| | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\|
| | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________
| | 9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | 9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\
| 3*|- - + ------------------------------------------------------------------------------|*im(t) \/ (273 - 288*re(t)) + 82944*im (t) *(-2 + 3*re(t))*sin|----------------------------------|| (-2 + 3*re(t))*|- - + ------------------------------------------------------------------------------| 3*\/ (273 - 288*re(t)) + 82944*im (t) *im(t)*sin|----------------------------------|
| \ 4 4 / \ 2 /| \ 4 4 / \ 2 /
x1 = I*|- ---------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------
| 2 2 / 2 2 \ | 2 2 / 2 2 \
\ (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/
$$x_{1} = \frac{\left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} - \frac{9}{4}\right) \left(3 \operatorname{re}{\left(t\right)} - 2\right)}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + i \left(- \frac{3 \left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} - \frac{9}{4}\right) \operatorname{im}{\left(t\right)}}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + \frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \left(3 \operatorname{re}{\left(t\right)} - 2\right) \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) + \frac{3 \sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)} \operatorname{im}{\left(t\right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}$$
/ / ___________________________________ \ \ / ___________________________________ \
| | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\|
| | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________
| |9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| |9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\
|3*|- + ------------------------------------------------------------------------------|*im(t) \/ (273 - 288*re(t)) + 82944*im (t) *(-2 + 3*re(t))*sin|----------------------------------|| (-2 + 3*re(t))*|- + ------------------------------------------------------------------------------| 3*\/ (273 - 288*re(t)) + 82944*im (t) *im(t)*sin|----------------------------------|
| \4 4 / \ 2 /| \4 4 / \ 2 /
x2 = I*|-------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------
| 2 2 / 2 2 \ | 2 2 / 2 2 \
\ (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/
$$x_{2} = - \frac{\left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} + \frac{9}{4}\right) \left(3 \operatorname{re}{\left(t\right)} - 2\right)}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + i \left(\frac{3 \left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} + \frac{9}{4}\right) \operatorname{im}{\left(t\right)}}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \left(3 \operatorname{re}{\left(t\right)} - 2\right) \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) - \frac{3 \sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)} \operatorname{im}{\left(t\right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}$$
x2 = -(((273 - 288*re(t))^2 + 82944*im(t)^2)^(1/4)*cos(atan2(-288*im(t, 273 - 288*re(t))/2)/4 + 9/4)*(3*re(t) - 2)/((3*re(t) - 2)^2 + 9*im(t)^2) + i*(3*(((273 - 288*re(t))^2 + 82944*im(t)^2)^(1/4)*cos(atan2(-288*im(t), 273 - 288*re(t))/2)/4 + 9/4)*im(t)/((3*re(t) - 2)^2 + 9*im(t)^2) - ((273 - 288*re(t))^2 + 82944*im(t)^2)^(1/4)*(3*re(t) - 2)*sin(atan2(-288*im(t), 273 - 288*re(t))/2)/(4*((3*re(t) - 2)^2 + 9*im(t)^2))) - 3*((273 - 288*re(t))^2 + 82944*im(t)^2)^(1/4)*sin(atan2(-288*im(t), 273 - 288*re(t))/2)*im(t)/(4*((3*re(t) - 2)^2 + 9*im(t)^2)))
Suma y producto de raíces
[src]
suma
/ / ___________________________________ \ \ / ___________________________________ \ / / ___________________________________ \ \ / ___________________________________ \ | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | 9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | 9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\ | |9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| |9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\ | 3*|- - + ------------------------------------------------------------------------------|*im(t) \/ (273 - 288*re(t)) + 82944*im (t) *(-2 + 3*re(t))*sin|----------------------------------|| (-2 + 3*re(t))*|- - + ------------------------------------------------------------------------------| 3*\/ (273 - 288*re(t)) + 82944*im (t) *im(t)*sin|----------------------------------| |3*|- + ------------------------------------------------------------------------------|*im(t) \/ (273 - 288*re(t)) + 82944*im (t) *(-2 + 3*re(t))*sin|----------------------------------|| (-2 + 3*re(t))*|- + ------------------------------------------------------------------------------| 3*\/ (273 - 288*re(t)) + 82944*im (t) *im(t)*sin|----------------------------------| | \ 4 4 / \ 2 /| \ 4 4 / \ 2 / | \4 4 / \ 2 /| \4 4 / \ 2 / I*|- ---------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------- + I*|-------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------- | 2 2 / 2 2 \ | 2 2 / 2 2 \ | 2 2 / 2 2 \ | 2 2 / 2 2 \ \ (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ \ (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/
$$\left(\frac{\left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} - \frac{9}{4}\right) \left(3 \operatorname{re}{\left(t\right)} - 2\right)}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + i \left(- \frac{3 \left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} - \frac{9}{4}\right) \operatorname{im}{\left(t\right)}}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + \frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \left(3 \operatorname{re}{\left(t\right)} - 2\right) \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) + \frac{3 \sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)} \operatorname{im}{\left(t\right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) + \left(- \frac{\left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} + \frac{9}{4}\right) \left(3 \operatorname{re}{\left(t\right)} - 2\right)}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + i \left(\frac{3 \left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} + \frac{9}{4}\right) \operatorname{im}{\left(t\right)}}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \left(3 \operatorname{re}{\left(t\right)} - 2\right) \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) - \frac{3 \sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)} \operatorname{im}{\left(t\right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right)$$
=
/ / ___________________________________ \ \ / / ___________________________________ \ \ / ___________________________________ \ / ___________________________________ \ | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| | | 9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | |9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | 9 \ 2 /| |9 \ 2 /| | 3*|- - + ------------------------------------------------------------------------------|*im(t) \/ (273 - 288*re(t)) + 82944*im (t) *(-2 + 3*re(t))*sin|----------------------------------|| |3*|- + ------------------------------------------------------------------------------|*im(t) \/ (273 - 288*re(t)) + 82944*im (t) *(-2 + 3*re(t))*sin|----------------------------------|| (-2 + 3*re(t))*|- - + ------------------------------------------------------------------------------| (-2 + 3*re(t))*|- + ------------------------------------------------------------------------------| | \ 4 4 / \ 2 /| | \4 4 / \ 2 /| \ 4 4 / \4 4 / I*|- ---------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------| + I*|-------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------- | 2 2 / 2 2 \ | | 2 2 / 2 2 \ | 2 2 2 2 \ (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / \ (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / (-2 + 3*re(t)) + 9*im (t) (-2 + 3*re(t)) + 9*im (t)
$$\frac{\left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} - \frac{9}{4}\right) \left(3 \operatorname{re}{\left(t\right)} - 2\right)}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} + \frac{9}{4}\right) \left(3 \operatorname{re}{\left(t\right)} - 2\right)}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + i \left(- \frac{3 \left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} - \frac{9}{4}\right) \operatorname{im}{\left(t\right)}}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + \frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \left(3 \operatorname{re}{\left(t\right)} - 2\right) \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) + i \left(\frac{3 \left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} + \frac{9}{4}\right) \operatorname{im}{\left(t\right)}}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \left(3 \operatorname{re}{\left(t\right)} - 2\right) \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right)$$
producto
/ / / ___________________________________ \ \ / ___________________________________ \ \ / / / ___________________________________ \ \ / ___________________________________ \ \ | | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | \/ (273 - 288*re(t)) + 82944*im (t) *cos|----------------------------------|| ___________________________________ | | | | 9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | 9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | |9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| |9 \ 2 /| 4 / 2 2 /atan2(-288*im(t), 273 - 288*re(t))\| | | 3*|- - + ------------------------------------------------------------------------------|*im(t) \/ (273 - 288*re(t)) + 82944*im (t) *(-2 + 3*re(t))*sin|----------------------------------|| (-2 + 3*re(t))*|- - + ------------------------------------------------------------------------------| 3*\/ (273 - 288*re(t)) + 82944*im (t) *im(t)*sin|----------------------------------|| | |3*|- + ------------------------------------------------------------------------------|*im(t) \/ (273 - 288*re(t)) + 82944*im (t) *(-2 + 3*re(t))*sin|----------------------------------|| (-2 + 3*re(t))*|- + ------------------------------------------------------------------------------| 3*\/ (273 - 288*re(t)) + 82944*im (t) *im(t)*sin|----------------------------------|| | | \ 4 4 / \ 2 /| \ 4 4 / \ 2 /| | | \4 4 / \ 2 /| \4 4 / \ 2 /| |I*|- ---------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------|*|I*|-------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------| - --------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------| | | 2 2 / 2 2 \ | 2 2 / 2 2 \ | | | 2 2 / 2 2 \ | 2 2 / 2 2 \ | \ \ (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / \ \ (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ / (-2 + 3*re(t)) + 9*im (t) 4*\(-2 + 3*re(t)) + 9*im (t)/ /
$$\left(\frac{\left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} - \frac{9}{4}\right) \left(3 \operatorname{re}{\left(t\right)} - 2\right)}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + i \left(- \frac{3 \left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} - \frac{9}{4}\right) \operatorname{im}{\left(t\right)}}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + \frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \left(3 \operatorname{re}{\left(t\right)} - 2\right) \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) + \frac{3 \sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)} \operatorname{im}{\left(t\right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) \left(- \frac{\left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} + \frac{9}{4}\right) \left(3 \operatorname{re}{\left(t\right)} - 2\right)}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} + i \left(\frac{3 \left(\frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4} + \frac{9}{4}\right) \operatorname{im}{\left(t\right)}}{\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}} - \frac{\sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \left(3 \operatorname{re}{\left(t\right)} - 2\right) \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right) - \frac{3 \sqrt[4]{\left(273 - 288 \operatorname{re}{\left(t\right)}\right)^{2} + 82944 \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{im}{\left(t\right)},273 - 288 \operatorname{re}{\left(t\right)} \right)}}{2} \right)} \operatorname{im}{\left(t\right)}}{4 \left(\left(3 \operatorname{re}{\left(t\right)} - 2\right)^{2} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2}\right)}\right)$$
=
-(12 - 18*re(t) + 18*I*im(t))
----------------------------------
2 2
4 - 12*re(t) + 9*im (t) + 9*re (t)
$$- \frac{- 18 \operatorname{re}{\left(t\right)} + 18 i \operatorname{im}{\left(t\right)} + 12}{9 \left(\operatorname{re}{\left(t\right)}\right)^{2} - 12 \operatorname{re}{\left(t\right)} + 9 \left(\operatorname{im}{\left(t\right)}\right)^{2} + 4}$$
-(12 - 18*re(t) + 18*i*im(t))/(4 - 12*re(t) + 9*im(t)^2 + 9*re(t)^2)