Sr Examen
3*(x-2)^2=5*(6-x)^2+(-3-y)^2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 2 3*(x - 2) = 5*(6 - x) + (-3 - y)
$$3 \left(x - 2\right)^{2} = 5 \left(6 - x\right)^{2} + \left(- y - 3\right)^{2}$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$3 \left(x - 2\right)^{2} = 5 \left(6 - x\right)^{2} + \left(- y - 3\right)^{2}$$
en
$$3 \left(x - 2\right)^{2} + \left(- 5 \left(6 - x\right)^{2} - \left(- y - 3\right)^{2}\right) = 0$$
Abramos la expresión en la ecuación
$$3 \left(x - 2\right)^{2} + \left(- 5 \left(6 - x\right)^{2} - \left(- y - 3\right)^{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- 2 x^{2} + 48 x - y^{2} - 6 y - 177 = 0$$
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
$$a = -2$$
$$b = 48$$
$$c = - y^{2} - 6 y - 177$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = 12 - \frac{\sqrt{- 8 y^{2} - 48 y + 888}}{4}$$
$$x_{2} = \frac{\sqrt{- 8 y^{2} - 48 y + 888}}{4} + 12$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$3 \left(x - 2\right)^{2} = 5 \left(6 - x\right)^{2} + \left(- y - 3\right)^{2}$$
en
$$3 \left(x - 2\right)^{2} + \left(- 5 \left(6 - x\right)^{2} - \left(- y - 3\right)^{2}\right) = 0$$
Abramos la expresión en la ecuación
$$3 \left(x - 2\right)^{2} + \left(- 5 \left(6 - x\right)^{2} - \left(- y - 3\right)^{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- 2 x^{2} + 48 x - y^{2} - 6 y - 177 = 0$$
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
$$a = -2$$
$$b = 48$$
$$c = - y^{2} - 6 y - 177$$
, entonces
D = b^2 - 4 * a * c =
(48)^2 - 4 * (-2) * (-177 - y^2 - 6*y) = 888 - 48*y - 8*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = 12 - \frac{\sqrt{- 8 y^{2} - 48 y + 888}}{4}$$
$$x_{2} = \frac{\sqrt{- 8 y^{2} - 48 y + 888}}{4} + 12$$
Gráfica
Respuesta rápida
[src]
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/|
\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|----------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|----------------------------------------------------------------------|
\ 2 / \ 2 /
x1 = 12 - -------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{1} = - \frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + 12$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/|
\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|----------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|----------------------------------------------------------------------|
\ 2 / \ 2 /
x2 = 12 + -------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{2} = \frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + 12$$
x2 = i*((-4*re(y)*im(y) - 12*im(y))^2 + (-2*re(y)^2 - 12*re(y) + 2*im(y)^2 + 222)^2)^(1/4)*sin(atan2(-4*re(y)*im(y) - 12*im(y, -2*re(y)^2 - 12*re(y) + 2*im(y)^2 + 222)/2)/2 + ((-4*re(y)*im(y) - 12*im(y))^2 + (-2*re(y)^2 - 12*re(y) + 2*im(y)^2 + 222)^2)^(1/4)*cos(atan2(-4*re(y)*im(y) - 12*im(y), -2*re(y)^2 - 12*re(y) + 2*im(y)^2 + 222)/2)/2 + 12)
Suma y producto de raíces
[src]
suma
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/|
\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|----------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|----------------------------------------------------------------------| \/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|----------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|----------------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
12 - -------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------- + 12 + -------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + 12\right) + \left(\frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + 12\right)$$
=
24
$$24$$
producto
/ ________________________________________________________________________ ________________________________________________________________________ \ / ________________________________________________________________________ ________________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/|| | 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/| 4 / 2 / 2 2 \ |atan2\-12*im(y) - 4*im(y)*re(y), 222 - 12*re(y) - 2*re (y) + 2*im (y)/|| | \/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|----------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|----------------------------------------------------------------------|| | \/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *cos|----------------------------------------------------------------------| I*\/ (-12*im(y) - 4*im(y)*re(y)) + \222 - 12*re(y) - 2*re (y) + 2*im (y)/ *sin|----------------------------------------------------------------------|| | \ 2 / \ 2 /| | \ 2 / \ 2 /| |12 - -------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------|*|12 + -------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------| \ 2 2 / \ 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + 12\right) \left(\frac{i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 12 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 12 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 222 \right)}}{2} \right)}}{2} + 12\right)$$
=
2 2 177 re (y) im (y) --- + ------ + 3*re(y) - ------ + 3*I*im(y) + I*im(y)*re(y) 2 2 2
$$\frac{\left(\operatorname{re}{\left(y\right)}\right)^{2}}{2} + i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 3 \operatorname{re}{\left(y\right)} - \frac{\left(\operatorname{im}{\left(y\right)}\right)^{2}}{2} + 3 i \operatorname{im}{\left(y\right)} + \frac{177}{2}$$
177/2 + re(y)^2/2 + 3*re(y) - im(y)^2/2 + 3*i*im(y) + i*im(y)*re(y)