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- La ecuación:
-
Ecuación x^4+2*x^2-8=0
-
Ecuación x*(x^2+8*x+16)=5*(x+4)
-
Ecuación -25-x=-17
-
Ecuación x^2+3x=4
- Expresar {x} en función de y en la ecuación:
- -18*x-2*y=9
- -14*x+10*y=19
- -9*x+1*y=3
- -17*x+1*y=-11
- Integral de d{x}:
- -7
- Forma canónica:
- -7
-
Expresiones idénticas
- x^ dos + dos *y*x-y^ dos =- siete
- x al cuadrado más 2 multiplicar por y multiplicar por x menos y al cuadrado es igual a menos 7
- x en el grado dos más dos multiplicar por y multiplicar por x menos y en el grado dos es igual a menos siete
- x2+2*y*x-y2=-7
- x²+2*y*x-y²=-7
- x en el grado 2+2*y*x-y en el grado 2=-7
- x^2+2yx-y^2=-7
- x2+2yx-y2=-7
-
Expresiones semejantes
- x^2-2*y*x-y^2=-7
- x^2+2*y*x-y^2=+7
- x^2+2*y*x+y^2=-7
x^2+2*y*x-y^2=-7 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$- y^{2} + \left(x^{2} + x 2 y\right) = -7$$
en
$$\left(- y^{2} + \left(x^{2} + x 2 y\right)\right) + 7 = 0$$
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
$$a = 1$$
$$b = 2 y$$
$$c = 7 - y^{2}$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = - y + \frac{\sqrt{8 y^{2} - 28}}{2}$$
$$x_{2} = - y - \frac{\sqrt{8 y^{2} - 28}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$- y^{2} + \left(x^{2} + x 2 y\right) = -7$$
en
$$\left(- y^{2} + \left(x^{2} + x 2 y\right)\right) + 7 = 0$$
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
$$a = 1$$
$$b = 2 y$$
$$c = 7 - y^{2}$$
, entonces
D = b^2 - 4 * a * c =
(2*y)^2 - 4 * (1) * (7 - y^2) = -28 + 8*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = - y + \frac{\sqrt{8 y^{2} - 28}}{2}$$
$$x_{2} = - y - \frac{\sqrt{8 y^{2} - 28}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2 y$$
$$q = \frac{c}{a}$$
$$q = 7 - y^{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - 2 y$$
$$x_{1} x_{2} = 7 - y^{2}$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2 y$$
$$q = \frac{c}{a}$$
$$q = 7 - y^{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - 2 y$$
$$x_{1} x_{2} = 7 - y^{2}$$
Gráfica
Respuesta rápida
[src]
/ ________________________________________________ \ ________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|
x1 = -re(y) + I*|-im(y) - \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *sin|----------------------------------------------|| - \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *cos|----------------------------------------------|
\ \ 2 // \ 2 /
$$x_{1} = i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}$$
/ ________________________________________________ \ ________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|
x2 = -re(y) + I*|-im(y) + \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *sin|----------------------------------------------|| + \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *cos|----------------------------------------------|
\ \ 2 // \ 2 /
$$x_{2} = i \left(\sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}$$
x2 = i*(((2*re(y)^2 - 2*im(y)^2 - 7)^2 + 16*re(y)^2*im(y)^2)^(1/4)*sin(atan2(4*re(y)*im(y, 2*re(y)^2 - 2*im(y)^2 - 7)/2) - im(y)) + ((2*re(y)^2 - 2*im(y)^2 - 7)^2 + 16*re(y)^2*im(y)^2)^(1/4)*cos(atan2(4*re(y)*im(y), 2*re(y)^2 - 2*im(y)^2 - 7)/2) - re(y))
Suma y producto de raíces
[src]
suma
/ ________________________________________________ \ ________________________________________________ / ________________________________________________ \ ________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\ | / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/| | 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|
-re(y) + I*|-im(y) - \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *sin|----------------------------------------------|| - \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *cos|----------------------------------------------| + -re(y) + I*|-im(y) + \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *sin|----------------------------------------------|| + \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *cos|----------------------------------------------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
$$\left(i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}\right) + \left(i \left(\sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}\right)$$
=
/ ________________________________________________ \ / ________________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| | 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||
-2*re(y) + I*|-im(y) + \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *sin|----------------------------------------------|| + I*|-im(y) - \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *sin|----------------------------------------------||
\ \ 2 // \ \ 2 //
$$i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + i \left(\sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - 2 \operatorname{re}{\left(y\right)}$$
producto
/ / ________________________________________________ \ ________________________________________________ \ / / ________________________________________________ \ ________________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| | | 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| 4 / / 2 2 \ 2 2 |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| |-re(y) + I*|-im(y) - \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *sin|----------------------------------------------|| - \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *cos|----------------------------------------------||*|-re(y) + I*|-im(y) + \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *sin|----------------------------------------------|| + \/ \-7 - 2*im (y) + 2*re (y)/ + 16*im (y)*re (y) *cos|----------------------------------------------|| \ \ \ 2 // \ 2 // \ \ \ 2 // \ 2 //
$$\left(i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}\right) \left(i \left(\sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}\right)$$
=
2 2 7 + im (y) - re (y) - 2*I*im(y)*re(y)
$$- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 7$$
7 + im(y)^2 - re(y)^2 - 2*i*im(y)*re(y)