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Expresiones semejantes
- 2*x^2+4*x*y-3y^2-16=0
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2*x^2-4*x*y-3y^2-16=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 2*x - 4*x*y - 3*y - 16 = 0
$$\left(- 3 y^{2} + \left(2 x^{2} - 4 x y\right)\right) - 16 = 0$$
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
$$a = 2$$
$$b = - 4 y$$
$$c = - 3 y^{2} - 16$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = y + \frac{\sqrt{40 y^{2} + 128}}{4}$$
$$x_{2} = y - \frac{\sqrt{40 y^{2} + 128}}{4}$$
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
$$a = 2$$
$$b = - 4 y$$
$$c = - 3 y^{2} - 16$$
, entonces
D = b^2 - 4 * a * c =
(-4*y)^2 - 4 * (2) * (-16 - 3*y^2) = 128 + 40*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = y + \frac{\sqrt{40 y^{2} + 128}}{4}$$
$$x_{2} = y - \frac{\sqrt{40 y^{2} + 128}}{4}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 3 y^{2} + \left(2 x^{2} - 4 x y\right)\right) - 16 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 2 x y - \frac{3 y^{2}}{2} - 8 = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - 2 y$$
$$q = \frac{c}{a}$$
$$q = - \frac{3 y^{2}}{2} - 8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 2 y$$
$$x_{1} x_{2} = - \frac{3 y^{2}}{2} - 8$$
$$\left(- 3 y^{2} + \left(2 x^{2} - 4 x y\right)\right) - 16 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 2 x y - \frac{3 y^{2}}{2} - 8 = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - 2 y$$
$$q = \frac{c}{a}$$
$$q = - \frac{3 y^{2}}{2} - 8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 2 y$$
$$x_{1} x_{2} = - \frac{3 y^{2}}{2} - 8$$
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\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| |4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *sin|-------------------------------------------------| | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *cos|-------------------------------------------------| |\/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *sin|-------------------------------------------------| | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *cos|-------------------------------------------------| | \ 2 / | \ 2 / | \ 2 / | \ 2 / I*|- -------------------------------------------------------------------------------------------------------------- + im(y)| - -------------------------------------------------------------------------------------------------------------- + re(y) + I*|-------------------------------------------------------------------------------------------------------------- + im(y)| + -------------------------------------------------------------------------------------------------------------- + re(y) \ 2 / 2 \ 2 / 2
$$\left(i \left(- \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) - \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)}\right) + \left(i \left(\frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)}\right)$$
=
/ ___________________________________________________ \ / ___________________________________________________ \
| / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ |
|4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | | 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| |
|\/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *sin|-------------------------------------------------| | | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *sin|-------------------------------------------------| |
| \ 2 / | | \ 2 / |
2*re(y) + I*|-------------------------------------------------------------------------------------------------------------- + im(y)| + I*|- -------------------------------------------------------------------------------------------------------------- + im(y)|
\ 2 / \ 2 /
$$i \left(- \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + i \left(\frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + 2 \operatorname{re}{\left(y\right)}$$
producto
/ / ___________________________________________________ \ ___________________________________________________ \ / / ___________________________________________________ \ ___________________________________________________ \ | | / 2 / / 2 2 \\ | / 2 / / 2 2 \\ | | | / 2 / / 2 2 \\ | / 2 / / 2 2 \\ | | | 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | | |4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | | | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *sin|-------------------------------------------------| | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *cos|-------------------------------------------------| | | |\/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *sin|-------------------------------------------------| | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *cos|-------------------------------------------------| | | | \ 2 / | \ 2 / | | | \ 2 / | \ 2 / | |I*|- -------------------------------------------------------------------------------------------------------------- + im(y)| - -------------------------------------------------------------------------------------------------------------- + re(y)|*|I*|-------------------------------------------------------------------------------------------------------------- + im(y)| + -------------------------------------------------------------------------------------------------------------- + re(y)| \ \ 2 / 2 / \ \ 2 / 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) - \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)}\right) \left(i \left(\frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)}\right)$$
=
2 2
3*re (y) 3*im (y)
-8 - -------- + -------- - 3*I*im(y)*re(y)
2 2
$$- \frac{3 \left(\operatorname{re}{\left(y\right)}\right)^{2}}{2} - 3 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + \frac{3 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{2} - 8$$
-8 - 3*re(y)^2/2 + 3*im(y)^2/2 - 3*i*im(y)*re(y)
Respuesta rápida
[src]
/ ___________________________________________________ \ ___________________________________________________
| / 2 / / 2 2 \\ | / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/|
| \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *sin|-------------------------------------------------| | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *cos|-------------------------------------------------|
| \ 2 / | \ 2 /
x1 = I*|- -------------------------------------------------------------------------------------------------------------- + im(y)| - -------------------------------------------------------------------------------------------------------------- + re(y)
\ 2 / 2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) - \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)}$$
/ ___________________________________________________ \ ___________________________________________________
| / 2 / / 2 2 \\ | / 2 / / 2 2 \\
|4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/| | 4 / / 2 2 \ 2 2 |atan2\20*im(y)*re(y), 32 - 10*im (y) + 10*re (y)/|
|\/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *sin|-------------------------------------------------| | \/ \32 - 10*im (y) + 10*re (y)/ + 400*im (y)*re (y) *cos|-------------------------------------------------|
| \ 2 / | \ 2 /
x2 = I*|-------------------------------------------------------------------------------------------------------------- + im(y)| + -------------------------------------------------------------------------------------------------------------- + re(y)
\ 2 / 2
$$x_{2} = i \left(\frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + \frac{\sqrt[4]{\left(10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32\right)^{2} + 400 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(20 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},10 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 10 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 32 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)}$$
x2 = i*(((10*re(y)^2 - 10*im(y)^2 + 32)^2 + 400*re(y)^2*im(y)^2)^(1/4)*sin(atan2(20*re(y)*im(y, 10*re(y)^2 - 10*im(y)^2 + 32)/2)/2 + im(y)) + ((10*re(y)^2 - 10*im(y)^2 + 32)^2 + 400*re(y)^2*im(y)^2)^(1/4)*cos(atan2(20*re(y)*im(y), 10*re(y)^2 - 10*im(y)^2 + 32)/2)/2 + re(y))