Sr Examen
Otras calculadoras
-
¿Cómo usar?
- La ecuación:
-
Ecuación x=(x+1)/2
- Ecuación (x-6)(-5x-9)=0
-
Ecuación x^2-14*x+33=0
-
Ecuación x^2+6=5x
- Expresar {x} en función de y en la ecuación:
- -13*x-19*y=17
- -9*x-12*y=-19
- -17*x-12*y=-9
- -8*x+3*y=-4
- Integral de d{x}:
- 19
- Suma de la serie:
-
19
- Límite de la función:
- 19
-
Expresiones idénticas
- x^ dos -8xy+15y^ dos = diecinueve
- x al cuadrado menos 8xy más 15y al cuadrado es igual a 19
- x en el grado dos menos 8xy más 15y en el grado dos es igual a diecinueve
- x2-8xy+15y2=19
- x²-8xy+15y²=19
- x en el grado 2-8xy+15y en el grado 2=19
-
Expresiones semejantes
- x^2+8xy+15y^2=19
- x^2-8xy-15y^2=19
x^2-8xy+15y^2=19 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
$$15 y^{2} + \left(x^{2} - 8 x y\right) = 19$$
en
$$\left(15 y^{2} + \left(x^{2} - 8 x y\right)\right) - 19 = 0$$
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
$$a = 1$$
$$b = - 8 y$$
$$c = 15 y^{2} - 19$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = 4 y + \frac{\sqrt{4 y^{2} + 76}}{2}$$
$$x_{2} = 4 y - \frac{\sqrt{4 y^{2} + 76}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$15 y^{2} + \left(x^{2} - 8 x y\right) = 19$$
en
$$\left(15 y^{2} + \left(x^{2} - 8 x y\right)\right) - 19 = 0$$
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
$$a = 1$$
$$b = - 8 y$$
$$c = 15 y^{2} - 19$$
, entonces
D = b^2 - 4 * a * c =
(-8*y)^2 - 4 * (1) * (-19 + 15*y^2) = 76 + 4*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = 4 y + \frac{\sqrt{4 y^{2} + 76}}{2}$$
$$x_{2} = 4 y - \frac{\sqrt{4 y^{2} + 76}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - 8 y$$
$$q = \frac{c}{a}$$
$$q = 15 y^{2} - 19$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8 y$$
$$x_{1} x_{2} = 15 y^{2} - 19$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - 8 y$$
$$q = \frac{c}{a}$$
$$q = 15 y^{2} - 19$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8 y$$
$$x_{1} x_{2} = 15 y^{2} - 19$$
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), 19 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/| | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|
4*re(y) + I*|4*im(y) - \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| - \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------| + 4*re(y) + I*|4*im(y) + \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| + \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
$$\left(i \left(- \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\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} + 19 \right)}}{2} \right)} + 4 \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\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} + 19 \right)}}{2} \right)} + 4 \operatorname{re}{\left(y\right)}\right) + \left(i \left(\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\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} + 19 \right)}}{2} \right)} + 4 \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\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} + 19 \right)}}{2} \right)} + 4 \operatorname{re}{\left(y\right)}\right)$$
=
/ ___________________________________________ \ / ___________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|| | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/||
8*re(y) + I*|4*im(y) + \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| + I*|4*im(y) - \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------||
\ \ 2 // \ \ 2 //
$$i \left(- \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19 \right)}}{2} \right)} + 4 \operatorname{im}{\left(y\right)}\right) + i \left(\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19 \right)}}{2} \right)} + 4 \operatorname{im}{\left(y\right)}\right) + 8 \operatorname{re}{\left(y\right)}$$
producto
/ / ___________________________________________ \ ___________________________________________ \ / / ___________________________________________ \ ___________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|| | | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|| |4*re(y) + I*|4*im(y) - \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| - \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------||*|4*re(y) + I*|4*im(y) + \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| + \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|| \ \ \ 2 // \ 2 // \ \ \ 2 // \ 2 //
$$\left(i \left(- \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\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} + 19 \right)}}{2} \right)} + 4 \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\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} + 19 \right)}}{2} \right)} + 4 \operatorname{re}{\left(y\right)}\right) \left(i \left(\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\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} + 19 \right)}}{2} \right)} + 4 \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\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} + 19 \right)}}{2} \right)} + 4 \operatorname{re}{\left(y\right)}\right)$$
=
2 2 -19 - 15*im (y) + 15*re (y) + 30*I*im(y)*re(y)
$$15 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 30 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 15 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 19$$
-19 - 15*im(y)^2 + 15*re(y)^2 + 30*i*im(y)*re(y)
Respuesta rápida
[src]
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|
x1 = 4*re(y) + I*|4*im(y) - \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| - \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|
\ \ 2 // \ 2 /
$$x_{1} = i \left(- \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19 \right)}}{2} \right)} + 4 \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19 \right)}}{2} \right)} + 4 \operatorname{re}{\left(y\right)}$$
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 19 + re (y) - im (y)/|
x2 = 4*re(y) + I*|4*im(y) + \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|------------------------------------------|| + \/ \19 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|------------------------------------------|
\ \ 2 // \ 2 /
$$x_{2} = i \left(\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19 \right)}}{2} \right)} + 4 \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 19 \right)}}{2} \right)} + 4 \operatorname{re}{\left(y\right)}$$
x2 = i*(((re(y)^2 - im(y)^2 + 19)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(2*re(y)*im(y, re(y)^2 - im(y)^2 + 19)/2) + 4*im(y)) + ((re(y)^2 - im(y)^2 + 19)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(2*re(y)*im(y), re(y)^2 - im(y)^2 + 19)/2) + 4*re(y))