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- La ecuación:
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Ecuación 2+x=6
-
Ecuación 53x^2=0
-
Ecuación (x+3)/(x-3)=(2*x+3)/x
- Ecuación 5*x+3*y=4
- Expresar {x} en función de y en la ecuación:
- 9*x+7*y=4
- -4*x+18*y=-6
- -10*x+15*y=-19
- 18*x-7*y=-13
- Integral de d{x}:
- -7
- Forma canónica:
- -7
-
Expresiones idénticas
- x^ dos -8x+y^ dos +2y=- siete
- x al cuadrado menos 8x más y al cuadrado más 2y es igual a menos 7
- x en el grado dos menos 8x más y en el grado dos más 2y es igual a menos siete
- x2-8x+y2+2y=-7
- x²-8x+y²+2y=-7
- x en el grado 2-8x+y en el grado 2+2y=-7
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Expresiones semejantes
- x^2-8x+y^2+2y=+7
- x^2-8x+y^2-2y=-7
- x^2+8x+y^2+2y=-7
- x^2-8x-y^2+2y=-7
x^2-8x+y^2+2y=-7 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 x - 8*x + y + 2*y = -7
$$2 y + \left(y^{2} + \left(x^{2} - 8 x\right)\right) = -7$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$2 y + \left(y^{2} + \left(x^{2} - 8 x\right)\right) = -7$$
en
$$\left(2 y + \left(y^{2} + \left(x^{2} - 8 x\right)\right)\right) + 7 = 0$$
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
$$a = 1$$
$$b = -8$$
$$c = y^{2} + 2 y + 7$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{- 4 y^{2} - 8 y + 36}}{2} + 4$$
$$x_{2} = 4 - \frac{\sqrt{- 4 y^{2} - 8 y + 36}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$2 y + \left(y^{2} + \left(x^{2} - 8 x\right)\right) = -7$$
en
$$\left(2 y + \left(y^{2} + \left(x^{2} - 8 x\right)\right)\right) + 7 = 0$$
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
$$a = 1$$
$$b = -8$$
$$c = y^{2} + 2 y + 7$$
, entonces
D = b^2 - 4 * a * c =
(-8)^2 - 4 * (1) * (7 + y^2 + 2*y) = 36 - 8*y - 4*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{- 4 y^{2} - 8 y + 36}}{2} + 4$$
$$x_{2} = 4 - \frac{\sqrt{- 4 y^{2} - 8 y + 36}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -8$$
$$q = \frac{c}{a}$$
$$q = y^{2} + 2 y + 7$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8$$
$$x_{1} x_{2} = y^{2} + 2 y + 7$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -8$$
$$q = \frac{c}{a}$$
$$q = y^{2} + 2 y + 7$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8$$
$$x_{1} x_{2} = y^{2} + 2 y + 7$$
Gráfica
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\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/|
4 - \/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *cos|--------------------------------------------------------------| - I*\/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *sin|--------------------------------------------------------------| + 4 + \/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *cos|--------------------------------------------------------------| + I*\/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + 4\right) + \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \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\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/|| | 4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/|| |4 - \/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *cos|--------------------------------------------------------------| - I*\/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *sin|--------------------------------------------------------------||*|4 + \/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *cos|--------------------------------------------------------------| + I*\/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *sin|--------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + 4\right) \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + 4\right)$$
=
2 2 7 + re (y) - im (y) + 2*re(y) + 2*I*im(y) + 2*I*im(y)*re(y)
$$\left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2 i \operatorname{im}{\left(y\right)} + 7$$
7 + re(y)^2 - im(y)^2 + 2*re(y) + 2*i*im(y) + 2*i*im(y)*re(y)
Respuesta rápida
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/|
x1 = 4 - \/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *cos|--------------------------------------------------------------| - I*\/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + 4$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) - 2*im(y)*re(y), 9 + im (y) - re (y) - 2*re(y)/|
x2 = 4 + \/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *cos|--------------------------------------------------------------| + I*\/ (-2*im(y) - 2*im(y)*re(y)) + \9 + im (y) - re (y) - 2*re(y)/ *sin|--------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + 4$$
x2 = i*((-2*re(y)*im(y) - 2*im(y))^2 + (-re(y)^2 - 2*re(y) + im(y)^2 + 9)^2)^(1/4)*sin(atan2(-2*re(y)*im(y) - 2*im(y, -re(y)^2 - 2*re(y) + im(y)^2 + 9)/2) + ((-2*re(y)*im(y) - 2*im(y))^2 + (-re(y)^2 - 2*re(y) + im(y)^2 + 9)^2)^(1/4)*cos(atan2(-2*re(y)*im(y) - 2*im(y), -re(y)^2 - 2*re(y) + im(y)^2 + 9)/2) + 4)