Sr Examen
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- La ecuación:
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Ecuación 2+x=6
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Ecuación 53x^2=0
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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
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Expresiones idénticas
- x^ dos + dos x+y^2-5y= cero
- x al cuadrado más 2x más y al cuadrado menos 5y es igual a 0
- x en el grado dos más dos x más y al cuadrado menos 5y es igual a cero
- x2+2x+y2-5y=0
- x²+2x+y²-5y=0
- x en el grado 2+2x+y en el grado 2-5y=0
- x^2+2x+y^2-5y=O
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Expresiones semejantes
- x^2+2x+y^2+5y=0
- x^2-2x+y^2-5y=0
- x^2+2x-y^2-5y=0
x^2+2x+y^2-5y=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 x + 2*x + y - 5*y = 0
$$- 5 y + \left(y^{2} + \left(x^{2} + 2 x\right)\right) = 0$$
Solución detallada
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
$$a = 1$$
$$b = 2$$
$$c = y^{2} - 5 y$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{- 4 y^{2} + 20 y + 4}}{2} - 1$$
$$x_{2} = - \frac{\sqrt{- 4 y^{2} + 20 y + 4}}{2} - 1$$
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
$$a = 1$$
$$b = 2$$
$$c = y^{2} - 5 y$$
, entonces
D = b^2 - 4 * a * c =
(2)^2 - 4 * (1) * (y^2 - 5*y) = 4 - 4*y^2 + 20*y
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{- 4 y^{2} + 20 y + 4}}{2} - 1$$
$$x_{2} = - \frac{\sqrt{- 4 y^{2} + 20 y + 4}}{2} - 1$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = y^{2} - 5 y$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -2$$
$$x_{1} x_{2} = y^{2} - 5 y$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = y^{2} - 5 y$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -2$$
$$x_{1} x_{2} = y^{2} - 5 y$$
Gráfica
Respuesta rápida
[src]
_______________________________________________________________ _______________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/| 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/|
x1 = -1 - \/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *cos|-------------------------------------------------------------| - I*\/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *sin|-------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - 1$$
_______________________________________________________________ _______________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/| 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/|
x2 = -1 + \/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *cos|-------------------------------------------------------------| + I*\/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *sin|-------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - 1$$
x2 = i*((-2*re(y)*im(y) + 5*im(y))^2 + (-re(y)^2 + 5*re(y) + im(y)^2 + 1)^2)^(1/4)*sin(atan2(-2*re(y)*im(y) + 5*im(y, -re(y)^2 + 5*re(y) + im(y)^2 + 1)/2) + ((-2*re(y)*im(y) + 5*im(y))^2 + (-re(y)^2 + 5*re(y) + im(y)^2 + 1)^2)^(1/4)*cos(atan2(-2*re(y)*im(y) + 5*im(y), -re(y)^2 + 5*re(y) + im(y)^2 + 1)/2) - 1)
Suma y producto de raíces
[src]
suma
_______________________________________________________________ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/| 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/| 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/| 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/|
-1 - \/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *cos|-------------------------------------------------------------| - I*\/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *sin|-------------------------------------------------------------| + -1 + \/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *cos|-------------------------------------------------------------| + I*\/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *sin|-------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - 1\right) + \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - 1\right)$$
=
-2
$$-2$$
producto
/ _______________________________________________________________ _______________________________________________________________ \ / _______________________________________________________________ _______________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/| 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/|| | 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/| 4 / 2 / 2 2 \ |atan2\5*im(y) - 2*im(y)*re(y), 1 + im (y) - re (y) + 5*re(y)/|| |-1 - \/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *cos|-------------------------------------------------------------| - I*\/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *sin|-------------------------------------------------------------||*|-1 + \/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *cos|-------------------------------------------------------------| + I*\/ (5*im(y) - 2*im(y)*re(y)) + \1 + im (y) - re (y) + 5*re(y)/ *sin|-------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - 1\right) \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 5 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 5 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - 1\right)$$
=
2 2 re (y) - im (y) - 5*re(y) - 5*I*im(y) + 2*I*im(y)*re(y)
$$\left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 5 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 5 i \operatorname{im}{\left(y\right)}$$
re(y)^2 - im(y)^2 - 5*re(y) - 5*i*im(y) + 2*i*im(y)*re(y)