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Expresiones semejantes
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((x+y)^2)-(8(x+y))-(2(x+y)x)-(4x)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 (x + y) - 8*(x + y) - 2*(x + y)*x - 4*x = 0
$$- 4 x + \left(- x 2 \left(x + y\right) + \left(\left(x + y\right)^{2} - 8 \left(x + y\right)\right)\right) = 0$$
Solución detallada
Abramos la expresión en la ecuación
$$- 4 x + \left(- x 2 \left(x + y\right) + \left(\left(x + y\right)^{2} - 8 \left(x + y\right)\right)\right) = 0$$
Obtenemos la ecuación cuadrática
$$- x^{2} - 12 x + y^{2} - 8 y = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = -8$$
$$c = - x^{2} - 12 x$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = \frac{\sqrt{4 x^{2} + 48 x + 64}}{2} + 4$$
$$y_{2} = 4 - \frac{\sqrt{4 x^{2} + 48 x + 64}}{2}$$
$$- 4 x + \left(- x 2 \left(x + y\right) + \left(\left(x + y\right)^{2} - 8 \left(x + y\right)\right)\right) = 0$$
Obtenemos la ecuación cuadrática
$$- x^{2} - 12 x + y^{2} - 8 y = 0$$
Es la ecuación de la forma
a*y^2 + b*y + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = -8$$
$$c = - x^{2} - 12 x$$
, entonces
D = b^2 - 4 * a * c =
(-8)^2 - 4 * (1) * (-x^2 - 12*x) = 64 + 4*x^2 + 48*x
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = \frac{\sqrt{4 x^{2} + 48 x + 64}}{2} + 4$$
$$y_{2} = 4 - \frac{\sqrt{4 x^{2} + 48 x + 64}}{2}$$
Gráfica
Respuesta rápida
[src]
__________________________________________________________________ __________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/| 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/|
y1 = 4 - \/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *cos|----------------------------------------------------------------| - I*\/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 /
$$y_{1} = - i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + 4$$
__________________________________________________________________ __________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/| 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/|
y2 = 4 + \/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *cos|----------------------------------------------------------------| + I*\/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 /
$$y_{2} = i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + 4$$
y2 = i*((2*re(x)*im(x) + 12*im(x))^2 + (re(x)^2 + 12*re(x) - im(x)^2 + 16)^2)^(1/4)*sin(atan2(2*re(x)*im(x) + 12*im(x, re(x)^2 + 12*re(x) - im(x)^2 + 16)/2) + ((2*re(x)*im(x) + 12*im(x))^2 + (re(x)^2 + 12*re(x) - im(x)^2 + 16)^2)^(1/4)*cos(atan2(2*re(x)*im(x) + 12*im(x), re(x)^2 + 12*re(x) - im(x)^2 + 16)/2) + 4)
Suma y producto de raíces
[src]
suma
__________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/| 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/| 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/| 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/|
4 - \/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *cos|----------------------------------------------------------------| - I*\/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *sin|----------------------------------------------------------------| + 4 + \/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *cos|----------------------------------------------------------------| + I*\/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + 4\right) + \left(i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + 4\right)$$
=
8
$$8$$
producto
/ __________________________________________________________________ __________________________________________________________________ \ / __________________________________________________________________ __________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/| 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/|| | 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/| 4 / 2 / 2 2 \ |atan2\12*im(x) + 2*im(x)*re(x), 16 + re (x) - im (x) + 12*re(x)/|| |4 - \/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *cos|----------------------------------------------------------------| - I*\/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *sin|----------------------------------------------------------------||*|4 + \/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *cos|----------------------------------------------------------------| + I*\/ (12*im(x) + 2*im(x)*re(x)) + \16 + re (x) - im (x) + 12*re(x)/ *sin|----------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + 4\right) \left(i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 12 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} + 12 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 16 \right)}}{2} \right)} + 4\right)$$
=
2 2 im (x) - re (x) - 12*re(x) - 12*I*im(x) - 2*I*im(x)*re(x)
$$- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 i \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 12 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} - 12 i \operatorname{im}{\left(x\right)}$$
im(x)^2 - re(x)^2 - 12*re(x) - 12*i*im(x) - 2*i*im(x)*re(x)