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- La ecuación:
-
Ecuación x^2=12
-
Ecuación -27x+220=-5x
-
Ecuación x^2+5x=36
-
Ecuación 3*x=8
- Expresar {x} en función de y en la ecuación:
- -17*x+2*y=9
- 10*x-15*y=16
- -17*x+2*y=-4
- -2*x-19*y=3
- Derivada de:
-
5^2
- Integral de d{x}:
- 5^2
- Forma canónica:
- 5^2
-
Expresiones idénticas
- (x+ dos)^ dos +(y- tres)^ dos = cinco ^ dos
- (x más 2) al cuadrado más (y menos 3) al cuadrado es igual a 5 al cuadrado
- (x más dos) en el grado dos más (y menos tres) en el grado dos es igual a cinco en el grado dos
- (x+2)2+(y-3)2=52
- x+22+y-32=52
- (x+2)²+(y-3)²=5²
- (x+2) en el grado 2+(y-3) en el grado 2=5 en el grado 2
- x+2^2+y-3^2=5^2
-
Expresiones semejantes
- (x+2)^2-(y-3)^2=5^2
- (x+2)^2+(y+3)^2=5^2
- (x-2)^2+(y-3)^2=5^2
(x+2)^2+(y-3)^2=5^2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 (x + 2) + (y - 3) = 25
$$\left(x + 2\right)^{2} + \left(y - 3\right)^{2} = 25$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(x + 2\right)^{2} + \left(y - 3\right)^{2} = 25$$
en
$$\left(\left(x + 2\right)^{2} + \left(y - 3\right)^{2}\right) - 25 = 0$$
Abramos la expresión en la ecuación
$$\left(\left(x + 2\right)^{2} + \left(y - 3\right)^{2}\right) - 25 = 0$$
Obtenemos la ecuación cuadrática
$$x^{2} + 4 x + y^{2} - 6 y - 12 = 0$$
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
$$a = 1$$
$$b = 4$$
$$c = y^{2} - 6 y - 12$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{- 4 y^{2} + 24 y + 64}}{2} - 2$$
$$x_{2} = - \frac{\sqrt{- 4 y^{2} + 24 y + 64}}{2} - 2$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(x + 2\right)^{2} + \left(y - 3\right)^{2} = 25$$
en
$$\left(\left(x + 2\right)^{2} + \left(y - 3\right)^{2}\right) - 25 = 0$$
Abramos la expresión en la ecuación
$$\left(\left(x + 2\right)^{2} + \left(y - 3\right)^{2}\right) - 25 = 0$$
Obtenemos la ecuación cuadrática
$$x^{2} + 4 x + y^{2} - 6 y - 12 = 0$$
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
$$a = 1$$
$$b = 4$$
$$c = y^{2} - 6 y - 12$$
, entonces
D = b^2 - 4 * a * c =
(4)^2 - 4 * (1) * (-12 + y^2 - 6*y) = 64 - 4*y^2 + 24*y
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{- 4 y^{2} + 24 y + 64}}{2} - 2$$
$$x_{2} = - \frac{\sqrt{- 4 y^{2} + 24 y + 64}}{2} - 2$$
Gráfica
Respuesta rápida
[src]
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/| 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/|
x1 = -2 - \/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *cos|--------------------------------------------------------------| - I*\/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - 2$$
________________________________________________________________ ________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/| 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/|
x2 = -2 + \/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *cos|--------------------------------------------------------------| + I*\/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - 2$$
x2 = i*((-2*re(y)*im(y) + 6*im(y))^2 + (-re(y)^2 + 6*re(y) + im(y)^2 + 16)^2)^(1/4)*sin(atan2(-2*re(y)*im(y) + 6*im(y, -re(y)^2 + 6*re(y) + im(y)^2 + 16)/2) + ((-2*re(y)*im(y) + 6*im(y))^2 + (-re(y)^2 + 6*re(y) + im(y)^2 + 16)^2)^(1/4)*cos(atan2(-2*re(y)*im(y) + 6*im(y), -re(y)^2 + 6*re(y) + im(y)^2 + 16)/2) - 2)
Suma y producto de raíces
[src]
suma
________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/| 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/| 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/| 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/|
-2 - \/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *cos|--------------------------------------------------------------| - I*\/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *sin|--------------------------------------------------------------| + -2 + \/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *cos|--------------------------------------------------------------| + I*\/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - 2\right) + \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - 2\right)$$
=
-4
$$-4$$
producto
/ ________________________________________________________________ ________________________________________________________________ \ / ________________________________________________________________ ________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/| 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/|| | 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/| 4 / 2 / 2 2 \ |atan2\6*im(y) - 2*im(y)*re(y), 16 + im (y) - re (y) + 6*re(y)/|| |-2 - \/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *cos|--------------------------------------------------------------| - I*\/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *sin|--------------------------------------------------------------||*|-2 + \/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *cos|--------------------------------------------------------------| + I*\/ (6*im(y) - 2*im(y)*re(y)) + \16 + im (y) - re (y) + 6*re(y)/ *sin|--------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - 2\right) \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 16 \right)}}{2} \right)} - 2\right)$$
=
2 2 -12 + re (y) - im (y) - 6*re(y) - 6*I*im(y) + 2*I*im(y)*re(y)
$$\left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 i \operatorname{im}{\left(y\right)} - 12$$
-12 + re(y)^2 - im(y)^2 - 6*re(y) - 6*i*im(y) + 2*i*im(y)*re(y)