Sr Examen
(x+2)²+(y–5)²=6 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 (x + 2) + (y - 5) = 6
$$\left(x + 2\right)^{2} + \left(y - 5\right)^{2} = 6$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(x + 2\right)^{2} + \left(y - 5\right)^{2} = 6$$
en
$$\left(\left(x + 2\right)^{2} + \left(y - 5\right)^{2}\right) - 6 = 0$$
Abramos la expresión en la ecuación
$$\left(\left(x + 2\right)^{2} + \left(y - 5\right)^{2}\right) - 6 = 0$$
Obtenemos la ecuación cuadrática
$$x^{2} + 4 x + y^{2} - 10 y + 23 = 0$$
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
$$a = 1$$
$$b = -10$$
$$c = x^{2} + 4 x + 23$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = \frac{\sqrt{- 4 x^{2} - 16 x + 8}}{2} + 5$$
$$y_{2} = 5 - \frac{\sqrt{- 4 x^{2} - 16 x + 8}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(x + 2\right)^{2} + \left(y - 5\right)^{2} = 6$$
en
$$\left(\left(x + 2\right)^{2} + \left(y - 5\right)^{2}\right) - 6 = 0$$
Abramos la expresión en la ecuación
$$\left(\left(x + 2\right)^{2} + \left(y - 5\right)^{2}\right) - 6 = 0$$
Obtenemos la ecuación cuadrática
$$x^{2} + 4 x + y^{2} - 10 y + 23 = 0$$
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
$$a = 1$$
$$b = -10$$
$$c = x^{2} + 4 x + 23$$
, entonces
D = b^2 - 4 * a * c =
(-10)^2 - 4 * (1) * (23 + x^2 + 4*x) = 8 - 16*x - 4*x^2
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = \frac{\sqrt{- 4 x^{2} - 16 x + 8}}{2} + 5$$
$$y_{2} = 5 - \frac{\sqrt{- 4 x^{2} - 16 x + 8}}{2}$$
Gráfica
Suma y producto de raíces
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suma
________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/| 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/| 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/| 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/|
5 - \/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *cos|--------------------------------------------------------------| - I*\/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *sin|--------------------------------------------------------------| + 5 + \/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *cos|--------------------------------------------------------------| + I*\/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + 5\right) + \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + 5\right)$$
=
10
$$10$$
producto
/ ________________________________________________________________ ________________________________________________________________ \ / ________________________________________________________________ ________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/| 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/| 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/|| |5 - \/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *cos|--------------------------------------------------------------| - I*\/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *sin|--------------------------------------------------------------||*|5 + \/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *cos|--------------------------------------------------------------| + I*\/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *sin|--------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + 5\right) \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + 5\right)$$
=
2 2 23 + re (x) - im (x) + 4*re(x) + 4*I*im(x) + 2*I*im(x)*re(x)
$$\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 i \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 4 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 i \operatorname{im}{\left(x\right)} + 23$$
23 + re(x)^2 - im(x)^2 + 4*re(x) + 4*i*im(x) + 2*i*im(x)*re(x)
Respuesta rápida
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/| 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/|
y1 = 5 - \/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *cos|--------------------------------------------------------------| - I*\/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 /
$$y_{1} = - i \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + 5$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/| 4 / 2 / 2 2 \ |atan2\-4*im(x) - 2*im(x)*re(x), 2 + im (x) - re (x) - 4*re(x)/|
y2 = 5 + \/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *cos|--------------------------------------------------------------| + I*\/ (-4*im(x) - 2*im(x)*re(x)) + \2 + im (x) - re (x) - 4*re(x)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 /
$$y_{2} = i \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 4 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2 \right)}}{2} \right)} + 5$$
y2 = i*((-2*re(x)*im(x) - 4*im(x))^2 + (-re(x)^2 - 4*re(x) + im(x)^2 + 2)^2)^(1/4)*sin(atan2(-2*re(x)*im(x) - 4*im(x, -re(x)^2 - 4*re(x) + im(x)^2 + 2)/2) + ((-2*re(x)*im(x) - 4*im(x))^2 + (-re(x)^2 - 4*re(x) + im(x)^2 + 2)^2)^(1/4)*cos(atan2(-2*re(x)*im(x) - 4*im(x), -re(x)^2 - 4*re(x) + im(x)^2 + 2)/2) + 5)