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(12-3x)^2+(2y-16)^6=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
          2             6    
(12 - 3*x)  + (2*y - 16)  = 0
$$\left(12 - 3 x\right)^{2} + \left(2 y - 16\right)^{6} = 0$$
Solución detallada
Abramos la expresión en la ecuación
$$\left(12 - 3 x\right)^{2} + \left(2 y - 16\right)^{6} = 0$$
Obtenemos la ecuación cuadrática
$$9 x^{2} - 72 x + 64 y^{6} - 3072 y^{5} + 61440 y^{4} - 655360 y^{3} + 3932160 y^{2} - 12582912 y + 16777360 = 0$$
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
$$a = 9$$
$$b = -72$$
$$c = 64 y^{6} - 3072 y^{5} + 61440 y^{4} - 655360 y^{3} + 3932160 y^{2} - 12582912 y + 16777360$$
, entonces
D = b^2 - 4 * a * c = 

(-72)^2 - 4 * (9) * (16777360 - 12582912*y - 655360*y^3 - 3072*y^5 + 64*y^6 + 61440*y^4 + 3932160*y^2) = -603979776 - 141557760*y^2 - 2211840*y^4 - 2304*y^6 + 110592*y^5 + 23592960*y^3 + 452984832*y

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

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

o
$$x_{1} = \frac{\sqrt{- 2304 y^{6} + 110592 y^{5} - 2211840 y^{4} + 23592960 y^{3} - 141557760 y^{2} + 452984832 y - 603979776}}{18} + 4$$
$$x_{2} = 4 - \frac{\sqrt{- 2304 y^{6} + 110592 y^{5} - 2211840 y^{4} + 23592960 y^{3} - 141557760 y^{2} + 452984832 y - 603979776}}{18}$$
Gráfica
Respuesta rápida [src]
             3        /                3                        \                        
         8*im (y)     |  8*(-8 + re(y))        2                |                 2      
x1 = 4 - -------- + I*|- --------------- + 8*im (y)*(-8 + re(y))| + 8*(-8 + re(y)) *im(y)
            3         \         3                               /                        
$$x_{1} = i \left(- \frac{8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{3}}{3} + 8 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) + 8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{2} \operatorname{im}{\left(y\right)} - \frac{8 \left(\operatorname{im}{\left(y\right)}\right)^{3}}{3} + 4$$
             3        /              3                        \                        
         8*im (y)     |8*(-8 + re(y))        2                |                 2      
x2 = 4 + -------- + I*|--------------- - 8*im (y)*(-8 + re(y))| - 8*(-8 + re(y)) *im(y)
            3         \       3                               /                        
$$x_{2} = i \left(\frac{8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{3}}{3} - 8 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) - 8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{2} \operatorname{im}{\left(y\right)} + \frac{8 \left(\operatorname{im}{\left(y\right)}\right)^{3}}{3} + 4$$
x2 = i*(8*(re(y) - 8)^3/3 - 8*(re(y) - 8)*im(y)^2) - 8*(re(y) - 8)^2*im(y) + 8*im(y)^3/3 + 4
Suma y producto de raíces [src]
suma
        3        /                3                        \                                   3        /              3                        \                        
    8*im (y)     |  8*(-8 + re(y))        2                |                 2             8*im (y)     |8*(-8 + re(y))        2                |                 2      
4 - -------- + I*|- --------------- + 8*im (y)*(-8 + re(y))| + 8*(-8 + re(y)) *im(y) + 4 + -------- + I*|--------------- - 8*im (y)*(-8 + re(y))| - 8*(-8 + re(y)) *im(y)
       3         \         3                               /                                  3         \       3                               /                        
$$\left(i \left(- \frac{8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{3}}{3} + 8 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) + 8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{2} \operatorname{im}{\left(y\right)} - \frac{8 \left(\operatorname{im}{\left(y\right)}\right)^{3}}{3} + 4\right) + \left(i \left(\frac{8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{3}}{3} - 8 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) - 8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{2} \operatorname{im}{\left(y\right)} + \frac{8 \left(\operatorname{im}{\left(y\right)}\right)^{3}}{3} + 4\right)$$
=
      /                3                        \     /              3                        \
      |  8*(-8 + re(y))        2                |     |8*(-8 + re(y))        2                |
8 + I*|- --------------- + 8*im (y)*(-8 + re(y))| + I*|--------------- - 8*im (y)*(-8 + re(y))|
      \         3                               /     \       3                               /
$$i \left(- \frac{8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{3}}{3} + 8 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) + i \left(\frac{8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{3}}{3} - 8 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) + 8$$
producto
/        3        /                3                        \                        \ /        3        /              3                        \                        \
|    8*im (y)     |  8*(-8 + re(y))        2                |                 2      | |    8*im (y)     |8*(-8 + re(y))        2                |                 2      |
|4 - -------- + I*|- --------------- + 8*im (y)*(-8 + re(y))| + 8*(-8 + re(y)) *im(y)|*|4 + -------- + I*|--------------- - 8*im (y)*(-8 + re(y))| - 8*(-8 + re(y)) *im(y)|
\       3         \         3                               /                        / \       3         \       3                               /                        /
$$\left(i \left(- \frac{8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{3}}{3} + 8 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) + 8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{2} \operatorname{im}{\left(y\right)} - \frac{8 \left(\operatorname{im}{\left(y\right)}\right)^{3}}{3} + 4\right) \left(i \left(\frac{8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{3}}{3} - 8 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) - 8 \left(\operatorname{re}{\left(y\right)} - 8\right)^{2} \operatorname{im}{\left(y\right)} + \frac{8 \left(\operatorname{im}{\left(y\right)}\right)^{3}}{3} + 4\right)$$
=
    /         3                    2             /            3       2                \\ /        3                    2             /            3       2                \\
-16*\-3 + 2*im (y) - 6*(-8 + re(y)) *im(y) + 2*I*\(-8 + re(y))  - 3*im (y)*(-8 + re(y))//*\3 + 2*im (y) - 6*(-8 + re(y)) *im(y) + 2*I*\(-8 + re(y))  - 3*im (y)*(-8 + re(y))//
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                                                                                      9                                                                                       
$$- \frac{16 \left(2 i \left(\left(\operatorname{re}{\left(y\right)} - 8\right)^{3} - 3 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) - 6 \left(\operatorname{re}{\left(y\right)} - 8\right)^{2} \operatorname{im}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{3} - 3\right) \left(2 i \left(\left(\operatorname{re}{\left(y\right)} - 8\right)^{3} - 3 \left(\operatorname{re}{\left(y\right)} - 8\right) \left(\operatorname{im}{\left(y\right)}\right)^{2}\right) - 6 \left(\operatorname{re}{\left(y\right)} - 8\right)^{2} \operatorname{im}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{3} + 3\right)}{9}$$
-16*(-3 + 2*im(y)^3 - 6*(-8 + re(y))^2*im(y) + 2*i*((-8 + re(y))^3 - 3*im(y)^2*(-8 + re(y))))*(3 + 2*im(y)^3 - 6*(-8 + re(y))^2*im(y) + 2*i*((-8 + re(y))^3 - 3*im(y)^2*(-8 + re(y))))/9