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- La ecuación:
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Ecuación x^3-x^2-x-1=0
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Ecuación 3x-2(3x+4)=10
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Ecuación 12-4(x-3)=39-9x
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Ecuación 0,2x+2,7=1,4-1,1x
- Expresar {x} en función de y en la ecuación:
- -11*x-3*y=14
- 2*x-15*y=-1
- -18*x-6*y=-1
- 6*x+9*y=-9
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Expresiones idénticas
- Xy= uno +x^(dos)
- Xy es igual a 1 más x en el grado (2)
- Xy es igual a uno más x en el grado (dos)
- Xy=1+x(2)
- Xy=1+x2
- Xy=1+x^2
-
Expresiones semejantes
- X*y=4
- Xy=1-x^(2)
Xy=1+x^(2) 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
$$x y = x^{2} + 1$$
en
$$x y + \left(- x^{2} - 1\right) = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = y$$
$$c = -1$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{y}{2} - \frac{\sqrt{y^{2} - 4}}{2}$$
$$x_{2} = \frac{y}{2} + \frac{\sqrt{y^{2} - 4}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$x y = x^{2} + 1$$
en
$$x y + \left(- x^{2} - 1\right) = 0$$
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
$$a = -1$$
$$b = y$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(y)^2 - 4 * (-1) * (-1) = -4 + y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{y}{2} - \frac{\sqrt{y^{2} - 4}}{2}$$
$$x_{2} = \frac{y}{2} + \frac{\sqrt{y^{2} - 4}}{2}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$x y = x^{2} + 1$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - x y + 1 = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - y$$
$$q = \frac{c}{a}$$
$$q = 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = y$$
$$x_{1} x_{2} = 1$$
$$x y = x^{2} + 1$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - x y + 1 = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - y$$
$$q = \frac{c}{a}$$
$$q = 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = y$$
$$x_{1} x_{2} = 1$$
Gráfica
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(y)*re(y), -4 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/| | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|
| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| | \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|
re(y) |im(y) \ 2 /| \ 2 / re(y) |im(y) \ 2 /| \ 2 /
----- + I*|----- - -----------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------- + ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
2 \ 2 2 / 2 2 \ 2 2 / 2
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}\right)$$
=
/ ___________________________________________ \ / ___________________________________________ \ | / 2 / / 2 2 \\| | / 2 / / 2 2 \\| | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| | \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| | \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| |im(y) \ 2 /| |im(y) \ 2 /| I*|----- + -----------------------------------------------------------------------------------------------| + I*|----- - -----------------------------------------------------------------------------------------------| + re(y) \ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \operatorname{re}{\left(y\right)}$$
producto
/ / ___________________________________________ \ ___________________________________________ \ / / ___________________________________________ \ ___________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| | | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| | | \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|| | | \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|| |re(y) |im(y) \ 2 /| \ 2 /| |re(y) |im(y) \ 2 /| \ 2 /| |----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------|*|----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------| \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}\right)$$
=
1
$$1$$
1
Respuesta rápida
[src]
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|
| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|
re(y) |im(y) \ 2 /| \ 2 /
x1 = ----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}$$
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|
| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| \/ \-4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|
re(y) |im(y) \ 2 /| \ 2 /
x2 = ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$x_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}$$
x2 = i*(((re(y)^2 - im(y)^2 - 4)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(2*re(y)*im(y, re(y)^2 - im(y)^2 - 4)/2)/2 + im(y)/2) + ((re(y)^2 - im(y)^2 - 4)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(2*re(y)*im(y), re(y)^2 - im(y)^2 - 4)/2)/2 + re(y)/2)