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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)- cuatro *x+y^(dos)+ tres = cero
- x en el grado (2) menos 4 multiplicar por x más y en el grado (2) más 3 es igual a 0
- x en el grado (dos) menos cuatro multiplicar por x más y en el grado (dos) más tres es igual a cero
- x(2)-4*x+y(2)+3=0
- x2-4*x+y2+3=0
- x^(2)-4x+y^(2)+3=0
- x(2)-4x+y(2)+3=0
- x2-4x+y2+3=0
- x^2-4x+y^2+3=0
- x^(2)-4*x+y^(2)+3=O
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Expresiones semejantes
- x^(2)-4*x-y^(2)+3=0
- x^(2)-4*x+y^(2)-3=0
- x^(2)+4*x+y^(2)+3=0
x^(2)-4*x+y^(2)+3=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 x - 4*x + y + 3 = 0
$$\left(y^{2} + \left(x^{2} - 4 x\right)\right) + 3 = 0$$
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
$$a = 1$$
$$b = -4$$
$$c = y^{2} + 3$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{4 - 4 y^{2}}}{2} + 2$$
$$x_{2} = 2 - \frac{\sqrt{4 - 4 y^{2}}}{2}$$
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
$$a = 1$$
$$b = -4$$
$$c = y^{2} + 3$$
, entonces
D = b^2 - 4 * a * c =
(-4)^2 - 4 * (1) * (3 + y^2) = 4 - 4*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{4 - 4 y^{2}}}{2} + 2$$
$$x_{2} = 2 - \frac{\sqrt{4 - 4 y^{2}}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = y^{2} + 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = y^{2} + 3$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = y^{2} + 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = y^{2} + 3$$
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), 1 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/|
x1 = 2 - \/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} + 2$$
__________________________________________ __________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/|
x2 = 2 + \/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1 \right)}}{2} \right)} + 2$$
x2 = i*((-re(y)^2 + im(y)^2 + 1)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(-2*re(y)*im(y, -re(y)^2 + im(y)^2 + 1)/2) + ((-re(y)^2 + im(y)^2 + 1)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(-2*re(y)*im(y), -re(y)^2 + im(y)^2 + 1)/2) + 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), 1 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/|
2 - \/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------| + 2 + \/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/ \1 + 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} + 1\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} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\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} + 1 \right)}}{2} \right)} + 2\right) + \left(i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\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} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\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} + 1 \right)}}{2} \right)} + 2\right)$$
=
4
$$4$$
producto
/ __________________________________________ __________________________________________ \ / __________________________________________ __________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/|| | 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/| 4 / / 2 2 \ 2 2 |atan2\-2*im(y)*re(y), 1 + im (y) - re (y)/|| |2 - \/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *sin|------------------------------------------||*|2 + \/ \1 + im (y) - re (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/ \1 + 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} + 1\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} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\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} + 1 \right)}}{2} \right)} + 2\right) \left(i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\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} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\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} + 1 \right)}}{2} \right)} + 2\right)$$
=
2 2 3 + re (y) - im (y) + 2*I*im(y)*re(y)
$$\left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 3$$
3 + re(y)^2 - im(y)^2 + 2*i*im(y)*re(y)