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- La ecuación:
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Ecuación x^4=(x-20)^2
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Ecuación 5-2*(x-1)=4-x
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Ecuación (x+6)^3=25*(x+6)
- Ecuación x+y+2=0
- Expresar {x} en función de y en la ecuación:
- 10*x-18*y=3
- 16*x+7*y=8
- -19*x+1*y=0
- -11*x-15*y=14
- Expresión:
- xy
- Forma canónica:
- xy
- Integral de d{x}:
- xy
- y^2+1
- Derivada de:
-
y^2+1
- Gráfico de la función y =:
-
y^2+1
-
Expresiones idénticas
- xy=y^ dos + uno
- xy es igual a y al cuadrado más 1
- xy es igual a y en el grado dos más uno
- xy=y2+1
- xy=y²+1
- xy=y en el grado 2+1
-
Expresiones semejantes
- xy=y^2-1
- 2*x^2-4*x*y-3y^2-16=0
- arcsin(x)*y=2^(x+y)-5
-
Expresiones con funciones
- xy
- xy-y^2=5x^2
- xy+lny-2cosx=0
- xy=-14
- xy^3+y^2=1
- xy+ln(y)=1
xy=y^2+1 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
$$x y = y^{2} + 1$$
en
$$x y + \left(- y^{2} - 1\right) = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = x$$
$$c = -1$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = \frac{x}{2} - \frac{\sqrt{x^{2} - 4}}{2}$$
$$y_{2} = \frac{x}{2} + \frac{\sqrt{x^{2} - 4}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$x y = y^{2} + 1$$
en
$$x y + \left(- y^{2} - 1\right) = 0$$
Es la ecuación de la forma
a*y^2 + b*y + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = x$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(x)^2 - 4 * (-1) * (-1) = -4 + x^2
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = \frac{x}{2} - \frac{\sqrt{x^{2} - 4}}{2}$$
$$y_{2} = \frac{x}{2} + \frac{\sqrt{x^{2} - 4}}{2}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$x y = y^{2} + 1$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$- x y + y^{2} + 1 = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - x$$
$$q = \frac{c}{a}$$
$$q = 1$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = x$$
$$y_{1} y_{2} = 1$$
$$x y = y^{2} + 1$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$- x y + y^{2} + 1 = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - x$$
$$q = \frac{c}{a}$$
$$q = 1$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = x$$
$$y_{1} y_{2} = 1$$
Gráfica
Respuesta rápida
[src]
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|
| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *sin|------------------------------------------|| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *cos|------------------------------------------|
re(x) |im(x) \ 2 /| \ 2 /
y1 = ----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$y_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}$$
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|
| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *sin|------------------------------------------|| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *cos|------------------------------------------|
re(x) |im(x) \ 2 /| \ 2 /
y2 = ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$y_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}$$
y2 = i*(((re(x)^2 - im(x)^2 - 4)^2 + 4*re(x)^2*im(x)^2)^(1/4)*sin(atan2(2*re(x)*im(x, re(x)^2 - im(x)^2 - 4)/2)/2 + im(x)/2) + ((re(x)^2 - im(x)^2 - 4)^2 + 4*re(x)^2*im(x)^2)^(1/4)*cos(atan2(2*re(x)*im(x), re(x)^2 - im(x)^2 - 4)/2)/2 + re(x)/2)
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\2*im(x)*re(x), -4 + re (x) - im (x)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/| | 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|
| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *sin|------------------------------------------|| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *cos|------------------------------------------| | \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *sin|------------------------------------------|| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *cos|------------------------------------------|
re(x) |im(x) \ 2 /| \ 2 / re(x) |im(x) \ 2 /| \ 2 /
----- + I*|----- - -----------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------- + ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
2 \ 2 2 / 2 2 \ 2 2 / 2
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}\right)$$
=
/ ___________________________________________ \ / ___________________________________________ \ | / 2 / / 2 2 \\| | / 2 / / 2 2 \\| | 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| | 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| | \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *sin|------------------------------------------|| | \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *sin|------------------------------------------|| |im(x) \ 2 /| |im(x) \ 2 /| I*|----- + -----------------------------------------------------------------------------------------------| + I*|----- - -----------------------------------------------------------------------------------------------| + re(x) \ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \operatorname{re}{\left(x\right)}$$
producto
/ / ___________________________________________ \ ___________________________________________ \ / / ___________________________________________ \ ___________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| | | 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| | | \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *sin|------------------------------------------|| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *cos|------------------------------------------|| | | \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *sin|------------------------------------------|| \/ \-4 + re (x) - im (x)/ + 4*im (x)*re (x) *cos|------------------------------------------|| |re(x) |im(x) \ 2 /| \ 2 /| |re(x) |im(x) \ 2 /| \ 2 /| |----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------|*|----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------| \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}\right)$$
=
1
$$1$$
1