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- La ecuación:
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Ecuación e^x=1
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Ecuación (x+9)^2=36*x
-
Ecuación a+120=2000/5
-
Ecuación x-32/4=50
- Expresar {x} en función de y en la ecuación:
- -6*x-12*y=9
- 9*x-1*y=17
- 10*x-2*y=11
- 12*x-3*y=-2
- Forma canónica:
- (x-2)^2+(y-1)^2
-
Expresiones idénticas
- (x- dos)^ dos +(y- uno)^ dos = nueve *(x+ cinco)^ dos
- (x menos 2) al cuadrado más (y menos 1) al cuadrado es igual a 9 multiplicar por (x más 5) al cuadrado
- (x menos dos) en el grado dos más (y menos uno) en el grado dos es igual a nueve multiplicar por (x más cinco) en el grado dos
- (x-2)2+(y-1)2=9*(x+5)2
- x-22+y-12=9*x+52
- (x-2)²+(y-1)²=9*(x+5)²
- (x-2) en el grado 2+(y-1) en el grado 2=9*(x+5) en el grado 2
- (x-2)^2+(y-1)^2=9(x+5)^2
- (x-2)2+(y-1)2=9(x+5)2
- x-22+y-12=9x+52
- x-2^2+y-1^2=9x+5^2
-
Expresiones semejantes
- (x-2)^2-(y-1)^2=9*(x+5)^2
- (x-2)^2+(y-1)^2=9*(x-5)^2
- (x+2)^2+(y-1)^2=9*(x+5)^2
- (x-2)^2+(y+1)^2=9*(x+5)^2
(x-2)^2+(y-1)^2=9*(x+5)^2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 2 (x - 2) + (y - 1) = 9*(x + 5)
$$\left(x - 2\right)^{2} + \left(y - 1\right)^{2} = 9 \left(x + 5\right)^{2}$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(x - 2\right)^{2} + \left(y - 1\right)^{2} = 9 \left(x + 5\right)^{2}$$
en
$$- 9 \left(x + 5\right)^{2} + \left(\left(x - 2\right)^{2} + \left(y - 1\right)^{2}\right) = 0$$
Abramos la expresión en la ecuación
$$- 9 \left(x + 5\right)^{2} + \left(\left(x - 2\right)^{2} + \left(y - 1\right)^{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- 8 x^{2} - 94 x + y^{2} - 2 y - 220 = 0$$
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
$$a = -8$$
$$b = -94$$
$$c = y^{2} - 2 y - 220$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = - \frac{\sqrt{32 y^{2} - 64 y + 1796}}{16} - \frac{47}{8}$$
$$x_{2} = \frac{\sqrt{32 y^{2} - 64 y + 1796}}{16} - \frac{47}{8}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(x - 2\right)^{2} + \left(y - 1\right)^{2} = 9 \left(x + 5\right)^{2}$$
en
$$- 9 \left(x + 5\right)^{2} + \left(\left(x - 2\right)^{2} + \left(y - 1\right)^{2}\right) = 0$$
Abramos la expresión en la ecuación
$$- 9 \left(x + 5\right)^{2} + \left(\left(x - 2\right)^{2} + \left(y - 1\right)^{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- 8 x^{2} - 94 x + y^{2} - 2 y - 220 = 0$$
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
$$a = -8$$
$$b = -94$$
$$c = y^{2} - 2 y - 220$$
, entonces
D = b^2 - 4 * a * c =
(-94)^2 - 4 * (-8) * (-220 + y^2 - 2*y) = 1796 - 64*y + 32*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = - \frac{\sqrt{32 y^{2} - 64 y + 1796}}{16} - \frac{47}{8}$$
$$x_{2} = \frac{\sqrt{32 y^{2} - 64 y + 1796}}{16} - \frac{47}{8}$$
Gráfica
Respuesta rápida
[src]
_________________________________________________________________________ _________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/| 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/|
\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *cos|-----------------------------------------------------------------------| I*\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *sin|-----------------------------------------------------------------------|
47 \ 2 / \ 2 /
x1 = - -- - ---------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------
8 8 8
$$x_{1} = - \frac{i \sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{47}{8}$$
_________________________________________________________________________ _________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/| 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/|
\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *cos|-----------------------------------------------------------------------| I*\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *sin|-----------------------------------------------------------------------|
47 \ 2 / \ 2 /
x2 = - -- + ---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------
8 8 8
$$x_{2} = \frac{i \sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{47}{8}$$
x2 = i*((16*re(y)*im(y) - 16*im(y))^2 + (8*re(y)^2 - 16*re(y) - 8*im(y)^2 + 449)^2)^(1/4)*sin(atan2(16*re(y)*im(y) - 16*im(y, 8*re(y)^2 - 16*re(y) - 8*im(y)^2 + 449)/2)/8 + ((16*re(y)*im(y) - 16*im(y))^2 + (8*re(y)^2 - 16*re(y) - 8*im(y)^2 + 449)^2)^(1/4)*cos(atan2(16*re(y)*im(y) - 16*im(y), 8*re(y)^2 - 16*re(y) - 8*im(y)^2 + 449)/2)/8 - 47/8)
Suma y producto de raíces
[src]
suma
_________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/| 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/| 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/| 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/|
\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *cos|-----------------------------------------------------------------------| I*\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *sin|-----------------------------------------------------------------------| \/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *cos|-----------------------------------------------------------------------| I*\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *sin|-----------------------------------------------------------------------|
47 \ 2 / \ 2 / 47 \ 2 / \ 2 /
- -- - ---------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------ + - -- + ---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------
8 8 8 8 8 8
$$\left(- \frac{i \sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{47}{8}\right) + \left(\frac{i \sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{47}{8}\right)$$
=
-47/4
$$- \frac{47}{4}$$
producto
/ _________________________________________________________________________ _________________________________________________________________________ \ / _________________________________________________________________________ _________________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/| 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/|| | 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/| 4 / 2 / 2 2 \ |atan2\-16*im(y) + 16*im(y)*re(y), 449 - 16*re(y) - 8*im (y) + 8*re (y)/|| | \/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *cos|-----------------------------------------------------------------------| I*\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *sin|-----------------------------------------------------------------------|| | \/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *cos|-----------------------------------------------------------------------| I*\/ (-16*im(y) + 16*im(y)*re(y)) + \449 - 16*re(y) - 8*im (y) + 8*re (y)/ *sin|-----------------------------------------------------------------------|| | 47 \ 2 / \ 2 /| | 47 \ 2 / \ 2 /| |- -- - ---------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------|*|- -- + ---------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------| \ 8 8 8 / \ 8 8 8 /
$$\left(- \frac{i \sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{47}{8}\right) \left(\frac{i \sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} + \frac{\sqrt[4]{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)}\right)^{2} + \left(8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(16 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 16 \operatorname{im}{\left(y\right)},8 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 16 \operatorname{re}{\left(y\right)} - 8 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 449 \right)}}{2} \right)}}{8} - \frac{47}{8}\right)$$
=
2 2 55 re (y) re(y) im (y) I*im(y) I*im(y)*re(y) -- - ------ + ----- + ------ + ------- - ------------- 2 8 4 8 4 4
$$- \frac{\left(\operatorname{re}{\left(y\right)}\right)^{2}}{8} - \frac{i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{4} + \frac{\operatorname{re}{\left(y\right)}}{4} + \frac{\left(\operatorname{im}{\left(y\right)}\right)^{2}}{8} + \frac{i \operatorname{im}{\left(y\right)}}{4} + \frac{55}{2}$$
55/2 - re(y)^2/8 + re(y)/4 + im(y)^2/8 + i*im(y)/4 - i*im(y)*re(y)/4