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- La ecuación:
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Ecuación 25*x^3-10*x^2+x=0
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Ecuación 4*(x-3)=x+6
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Ecuación -x-7=x
-
Ecuación 9-4x=3x-40
- Expresar {x} en función de y en la ecuación:
- 5*x+4*y=-12
- 5*x-13*y=17
- 4*x-11*y=-2
- 3*x-7*y=13
- Factorizar el polinomio:
- x^2+y^2
- Forma canónica:
- x^2+y^2
- Integral de d{x}:
- x^2+y^2
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Expresiones idénticas
- x^ dos +y^ dos = nueve
- x al cuadrado más y al cuadrado es igual a 9
- x en el grado dos más y en el grado dos es igual a nueve
- x2+y2=9
- x²+y²=9
- x en el grado 2+y en el grado 2=9
-
Expresiones semejantes
- x^2-y^2=9
x^2+y^2=9 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
$$x^{2} + y^{2} = 9$$
en
$$\left(x^{2} + y^{2}\right) - 9 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = y^{2} - 9$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{36 - 4 y^{2}}}{2}$$
$$x_{2} = - \frac{\sqrt{36 - 4 y^{2}}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$x^{2} + y^{2} = 9$$
en
$$\left(x^{2} + y^{2}\right) - 9 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = y^{2} - 9$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (-9 + y^2) = 36 - 4*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{36 - 4 y^{2}}}{2}$$
$$x_{2} = - \frac{\sqrt{36 - 4 y^{2}}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = y^{2} - 9$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = y^{2} - 9$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = y^{2} - 9$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = y^{2} - 9$$
Gráfica
Respuesta rápida
[src]
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/|
x1 = - \/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)}$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/|
x2 = \/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}{2} \right)}$$
x2 = i*((-re(y)^2 + im(y)^2 + 9)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(-2*re(y)*im(y, -re(y)^2 + im(y)^2 + 9)/2) + ((-re(y)^2 + im(y)^2 + 9)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(-2*re(y)*im(y), -re(y)^2 + im(y)^2 + 9)/2))
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 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/|
- \/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------| + \/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\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} + 9 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\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} + 9 \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\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} + 9 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\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} + 9 \right)}}{2} \right)}\right)$$
=
0
$$0$$
producto
/ __________________________________________ __________________________________________ \ / __________________________________________ __________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/|| |4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/|| |- \/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------||*|\/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\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} + 9 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\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} + 9 \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\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} + 9 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\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} + 9 \right)}}{2} \right)}\right)$$
=
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/ 2 / 2 2 \
/ / 2 2 \ 2 2 I*atan2\-2*im(y)*re(y), 9 + im (y) - re (y)/
-\/ \9 + im (y) - re (y)/ + 4*im (y)*re (y) *e
$$- \sqrt{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 9 \right)}}$$
-sqrt((9 + im(y)^2 - re(y)^2)^2 + 4*im(y)^2*re(y)^2)*exp(i*atan2(-2*im(y)*re(y), 9 + im(y)^2 - re(y)^2))