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2*y^2-8*y+3*a*x^2+12*x-2*x*y*a-3*x*y=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2              2                             
2*y  - 8*y + 3*a*x  + 12*x - 2*x*y*a - 3*x*y = 0
−3xy+(−a2xy+(12x+(3ax2+(2y2−8y))))=0- 3 x y + \left(- a 2 x y + \left(12 x + \left(3 a x^{2} + \left(2 y^{2} - 8 y\right)\right)\right)\right) = 0
Simplificar
Solución detallada
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:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=3aa = 3 a
b=−2ay−3y+12b = - 2 a y - 3 y + 12
c=2y2−8yc = 2 y^{2} - 8 y
, entonces
D = b^2 - 4 * a * c = 

(12 - 3*y - 2*a*y)^2 - 4 * (3*a) * (-8*y + 2*y^2) = (12 - 3*y - 2*a*y)^2 - 12*a*(-8*y + 2*y^2)

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
x1=2ay+3y+−12a(2y2−8y)+(−2ay−3y+12)2−126ax_{1} = \frac{2 a y + 3 y + \sqrt{- 12 a \left(2 y^{2} - 8 y\right) + \left(- 2 a y - 3 y + 12\right)^{2}} - 12}{6 a}
x2=2ay+3y−−12a(2y2−8y)+(−2ay−3y+12)2−126ax_{2} = \frac{2 a y + 3 y - \sqrt{- 12 a \left(2 y^{2} - 8 y\right) + \left(- 2 a y - 3 y + 12\right)^{2}} - 12}{6 a}
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
3ax2−2axy−3xy+12x+2y2−8y=03 a x^{2} - 2 a x y - 3 x y + 12 x + 2 y^{2} - 8 y = 0
Коэффициент при x равен
3a3 a
entonces son posibles los casos para a :
a<0a < 0
a=0a = 0
Consideremos todos los casos con detalles:
Con
a<0a < 0
la ecuación será
−3x2−xy+12x+2y2−8y=0- 3 x^{2} - x y + 12 x + 2 y^{2} - 8 y = 0
su solución
x=2y3x = \frac{2 y}{3}
x=4−yx = 4 - y
Con
a=0a = 0
la ecuación será
−3xy+12x+2y2−8y=0- 3 x y + 12 x + 2 y^{2} - 8 y = 0
su solución
x=2y3x = \frac{2 y}{3}
Teorema de Cardano-Vieta
reescribamos la ecuación
−3xy+(−a2xy+(12x+(3ax2+(2y2−8y))))=0- 3 x y + \left(- a 2 x y + \left(12 x + \left(3 a x^{2} + \left(2 y^{2} - 8 y\right)\right)\right)\right) = 0
de
ax2+bx+c=0a x^{2} + b x + c = 0
como ecuación cuadrática reducida
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
3ax2−2axy−3xy+12x+2y2−8y3a=0\frac{3 a x^{2} - 2 a x y - 3 x y + 12 x + 2 y^{2} - 8 y}{3 a} = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=−2ay−3y+123ap = \frac{- 2 a y - 3 y + 12}{3 a}
q=caq = \frac{c}{a}
q=2y2−8y3aq = \frac{2 y^{2} - 8 y}{3 a}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−−2ay−3y+123ax_{1} + x_{2} = - \frac{- 2 a y - 3 y + 12}{3 a}
x1x2=2y2−8y3ax_{1} x_{2} = \frac{2 y^{2} - 8 y}{3 a}
Gráfica
Suma y producto de raíces [src] 
suma
2*re(y)   2*I*im(y)     /  im(y)*re(a)     (-4 + re(y))*im(a)\   (-4 + re(y))*re(a)     im(a)*im(y)  
------- + --------- + I*|--------------- - ------------------| + ------------------ + ---------------
   3          3         |  2        2         2        2     |      2        2          2        2   
                        \im (a) + re (a)    im (a) + re (a)  /    im (a) + re (a)     im (a) + re (a)
(2re⁡(y)3+2iim⁡(y)3)+(i(−(re⁡(y)−4)im⁡(a)(re⁡(a))2+(im⁡(a))2+re⁡(a)im⁡(y)(re⁡(a))2+(im⁡(a))2)+(re⁡(y)−4)re⁡(a)(re⁡(a))2+(im⁡(a))2+im⁡(a)im⁡(y)(re⁡(a))2+(im⁡(a))2)\left(\frac{2 \operatorname{re}{\left(y\right)}}{3} + \frac{2 i \operatorname{im}{\left(y\right)}}{3}\right) + \left(i \left(- \frac{\left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) 
=
2*re(y)     /  im(y)*re(a)     (-4 + re(y))*im(a)\   2*I*im(y)   (-4 + re(y))*re(a)     im(a)*im(y)  
------- + I*|--------------- - ------------------| + --------- + ------------------ + ---------------
   3        |  2        2         2        2     |       3          2        2          2        2   
            \im (a) + re (a)    im (a) + re (a)  /                im (a) + re (a)     im (a) + re (a)
i(−(re⁡(y)−4)im⁡(a)(re⁡(a))2+(im⁡(a))2+re⁡(a)im⁡(y)(re⁡(a))2+(im⁡(a))2)+2re⁡(y)3+2iim⁡(y)3+(re⁡(y)−4)re⁡(a)(re⁡(a))2+(im⁡(a))2+im⁡(a)im⁡(y)(re⁡(a))2+(im⁡(a))2i \left(- \frac{\left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{2 \operatorname{re}{\left(y\right)}}{3} + \frac{2 i \operatorname{im}{\left(y\right)}}{3} + \frac{\left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} 
producto
/2*re(y)   2*I*im(y)\ /  /  im(y)*re(a)     (-4 + re(y))*im(a)\   (-4 + re(y))*re(a)     im(a)*im(y)  \
|------- + ---------|*|I*|--------------- - ------------------| + ------------------ + ---------------|
\   3          3    / |  |  2        2         2        2     |      2        2          2        2   |
                      \  \im (a) + re (a)    im (a) + re (a)  /    im (a) + re (a)     im (a) + re (a)/
(2re⁡(y)3+2iim⁡(y)3)(i(−(re⁡(y)−4)im⁡(a)(re⁡(a))2+(im⁡(a))2+re⁡(a)im⁡(y)(re⁡(a))2+(im⁡(a))2)+(re⁡(y)−4)re⁡(a)(re⁡(a))2+(im⁡(a))2+im⁡(a)im⁡(y)(re⁡(a))2+(im⁡(a))2)\left(\frac{2 \operatorname{re}{\left(y\right)}}{3} + \frac{2 i \operatorname{im}{\left(y\right)}}{3}\right) \left(i \left(- \frac{\left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) 
=
2*(I*im(y) + re(y))*(I*(im(y)*re(a) - (-4 + re(y))*im(a)) + (-4 + re(y))*re(a) + im(a)*im(y))
---------------------------------------------------------------------------------------------
                                       /  2        2   \                                     
                                     3*\im (a) + re (a)/
2(re⁡(y)+iim⁡(y))(i(−(re⁡(y)−4)im⁡(a)+re⁡(a)im⁡(y))+(re⁡(y)−4)re⁡(a)+im⁡(a)im⁡(y))3((re⁡(a))2+(im⁡(a))2)\frac{2 \left(\operatorname{re}{\left(y\right)} + i \operatorname{im}{\left(y\right)}\right) \left(i \left(- \left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}\right) + \left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{re}{\left(a\right)} + \operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}\right)}{3 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} 
2*(i*im(y) + re(y))*(i*(im(y)*re(a) - (-4 + re(y))*im(a)) + (-4 + re(y))*re(a) + im(a)*im(y))/(3*(im(a)^2 + re(a)^2))
Respuesta rápida [src] 
     2*re(y)   2*I*im(y)
x1 = ------- + ---------
        3          3
x1=2re⁡(y)3+2iim⁡(y)3x_{1} = \frac{2 \operatorname{re}{\left(y\right)}}{3} + \frac{2 i \operatorname{im}{\left(y\right)}}{3} 
       /  im(y)*re(a)     (-4 + re(y))*im(a)\   (-4 + re(y))*re(a)     im(a)*im(y)  
x2 = I*|--------------- - ------------------| + ------------------ + ---------------
       |  2        2         2        2     |      2        2          2        2   
       \im (a) + re (a)    im (a) + re (a)  /    im (a) + re (a)     im (a) + re (a)
x2=i(−(re⁡(y)−4)im⁡(a)(re⁡(a))2+(im⁡(a))2+re⁡(a)im⁡(y)(re⁡(a))2+(im⁡(a))2)+(re⁡(y)−4)re⁡(a)(re⁡(a))2+(im⁡(a))2+im⁡(a)im⁡(y)(re⁡(a))2+(im⁡(a))2x_{2} = i \left(- \frac{\left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{re}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(y\right)} - 4\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\operatorname{im}{\left(a\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} 
x2 = i*(-(re(y) - 4)*im(a)/(re(a)^2 + im(a)^2) + re(a)*im(y)/(re(a)^2 + im(a)^2)) + (re(y) - 4)*re(a)/(re(a)^2 + im(a)^2) + im(a)*im(y)/(re(a)^2 + im(a)^2)
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