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- La ecuación:
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Ecuación 2+x=6
-
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
- Derivada de:
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-3
- Suma de la serie:
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-3
- Integral de d{x}:
- -3
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Expresiones idénticas
- x^ dos -7x+y^ dos =- tres
- x al cuadrado menos 7x más y al cuadrado es igual a menos 3
- x en el grado dos menos 7x más y en el grado dos es igual a menos tres
- x2-7x+y2=-3
- x²-7x+y²=-3
- x en el grado 2-7x+y en el grado 2=-3
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Expresiones semejantes
- x^2-7x+y^2=+3
- x^2-7x-y^2=-3
- x^2+7x+y^2=-3
x^2-7x+y^2=-3 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
$$y^{2} + \left(x^{2} - 7 x\right) = -3$$
en
$$\left(y^{2} + \left(x^{2} - 7 x\right)\right) + 3 = 0$$
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
$$a = 1$$
$$b = -7$$
$$c = y^{2} + 3$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{37 - 4 y^{2}}}{2} + \frac{7}{2}$$
$$x_{2} = \frac{7}{2} - \frac{\sqrt{37 - 4 y^{2}}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$y^{2} + \left(x^{2} - 7 x\right) = -3$$
en
$$\left(y^{2} + \left(x^{2} - 7 x\right)\right) + 3 = 0$$
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
$$a = 1$$
$$b = -7$$
$$c = y^{2} + 3$$
, entonces
D = b^2 - 4 * a * c =
(-7)^2 - 4 * (1) * (3 + y^2) = 37 - 4*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{37 - 4 y^{2}}}{2} + \frac{7}{2}$$
$$x_{2} = \frac{7}{2} - \frac{\sqrt{37 - 4 y^{2}}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = y^{2} + 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 7$$
$$x_{1} x_{2} = y^{2} + 3$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = y^{2} + 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 7$$
$$x_{1} x_{2} = y^{2} + 3$$
Gráfica
Respuesta rápida
[src]
________________________________________________ ________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/| 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/|
\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *cos|-----------------------------------------------| I*\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *sin|-----------------------------------------------|
7 \ 2 / \ 2 /
x1 = - - --------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------
2 2 2
$$x_{1} = - \frac{i \sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{7}{2}$$
________________________________________________ ________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/| 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/|
\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *cos|-----------------------------------------------| I*\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *sin|-----------------------------------------------|
7 \ 2 / \ 2 /
x2 = - + --------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------------
2 2 2
$$x_{2} = \frac{i \sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{7}{2}$$
x2 = i*((-4*re(y)^2 + 4*im(y)^2 + 37)^2 + 64*re(y)^2*im(y)^2)^(1/4)*sin(atan2(-8*re(y)*im(y, -4*re(y)^2 + 4*im(y)^2 + 37)/2)/2 + ((-4*re(y)^2 + 4*im(y)^2 + 37)^2 + 64*re(y)^2*im(y)^2)^(1/4)*cos(atan2(-8*re(y)*im(y), -4*re(y)^2 + 4*im(y)^2 + 37)/2)/2 + 7/2)
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\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/| 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/| 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/| 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/|
\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *cos|-----------------------------------------------| I*\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *sin|-----------------------------------------------| \/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *cos|-----------------------------------------------| I*\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *sin|-----------------------------------------------|
7 \ 2 / \ 2 / 7 \ 2 / \ 2 /
- - --------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------- + - + --------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------------
2 2 2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{7}{2}\right) + \left(\frac{i \sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{7}{2}\right)$$
=
7
$$7$$
producto
/ ________________________________________________ ________________________________________________ \ / ________________________________________________ ________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/| 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/|| | 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/| 4 / / 2 2 \ 2 2 |atan2\-8*im(y)*re(y), 37 - 4*re (y) + 4*im (y)/|| | \/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *cos|-----------------------------------------------| I*\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *sin|-----------------------------------------------|| | \/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *cos|-----------------------------------------------| I*\/ \37 - 4*re (y) + 4*im (y)/ + 64*im (y)*re (y) *sin|-----------------------------------------------|| |7 \ 2 / \ 2 /| |7 \ 2 / \ 2 /| |- - --------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------|*|- + --------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------------| \2 2 2 / \2 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{7}{2}\right) \left(\frac{i \sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37\right)^{2} + 64 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 8 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 37 \right)}}{2} \right)}}{2} + \frac{7}{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)