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Expresar x en función de y en la ecuación x^2*y^2+x^2+y^2-14*x*y+2x-2y+37=0

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Solución numérica:

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Solución

Ha introducido [src] 
 2  2    2    2                              
x *y  + x  + y  - 14*x*y + 2*x - 2*y + 37 = 0
(−2y+(2x+(−14xy+(y2+(x2y2+x2)))))+37=0\left(- 2 y + \left(2 x + \left(- 14 x y + \left(y^{2} + \left(x^{2} y^{2} + x^{2}\right)\right)\right)\right)\right) + 37 = 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=y2+1a = y^{2} + 1
b=2−14yb = 2 - 14 y
c=y2−2y+37c = y^{2} - 2 y + 37
, entonces
D = b^2 - 4 * a * c = 

(2 - 14*y)^2 - 4 * (1 + y^2) * (37 + y^2 - 2*y) = (2 - 14*y)^2 - (4 + 4*y^2)*(37 + y^2 - 2*y)

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

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

o
x1=14y+(2−14y)2−(4y2+4)(y2−2y+37)−22y2+2x_{1} = \frac{14 y + \sqrt{\left(2 - 14 y\right)^{2} - \left(4 y^{2} + 4\right) \left(y^{2} - 2 y + 37\right)} - 2}{2 y^{2} + 2}
x2=14y−(2−14y)2−(4y2+4)(y2−2y+37)−22y2+2x_{2} = \frac{14 y - \sqrt{\left(2 - 14 y\right)^{2} - \left(4 y^{2} + 4\right) \left(y^{2} - 2 y + 37\right)} - 2}{2 y^{2} + 2}
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
x2y2+x2−14xy+2x+y2−2y+37=0x^{2} y^{2} + x^{2} - 14 x y + 2 x + y^{2} - 2 y + 37 = 0
Коэффициент при x равен
y2+1y^{2} + 1
entonces son posibles los casos para y :
Consideremos todos los casos con detalles:
Respuesta rápida [src] 
       //      2        2   \ /      2        2                     \                                                       \   /      2        2   \                                            /      2        2                     \            
       |\1 + re (y) - im (y)/*\6 + im (y) - re (y) + 7*im(y) + re(y)/   2*(-1 - im(y) + 7*re(y) + 2*im(y)*re(y))*im(y)*re(y)|   \1 + re (y) - im (y)/*(-1 - im(y) + 7*re(y) + 2*im(y)*re(y))   2*\6 + im (y) - re (y) + 7*im(y) + re(y)/*im(y)*re(y)
x1 = I*|------------------------------------------------------------- - ----------------------------------------------------| + ------------------------------------------------------------ + -----------------------------------------------------
       |                                2                                                          2                        |                                  2                                                           2                        
       |           /      2        2   \        2      2                      /      2        2   \        2      2         |             /      2        2   \        2      2                       /      2        2   \        2      2         
       \           \1 + re (y) - im (y)/  + 4*im (y)*re (y)                   \1 + re (y) - im (y)/  + 4*im (y)*re (y)      /             \1 + re (y) - im (y)/  + 4*im (y)*re (y)                    \1 + re (y) - im (y)/  + 4*im (y)*re (y)
x1=i(((re⁡(y))2−(im⁡(y))2+1)(−(re⁡(y))2+re⁡(y)+(im⁡(y))2+7im⁡(y)+6)((re⁡(y))2−(im⁡(y))2+1)2+4(re⁡(y))2(im⁡(y))2−2(2re⁡(y)im⁡(y)+7re⁡(y)−im⁡(y)−1)re⁡(y)im⁡(y)((re⁡(y))2−(im⁡(y))2+1)2+4(re⁡(y))2(im⁡(y))2)+((re⁡(y))2−(im⁡(y))2+1)(2re⁡(y)im⁡(y)+7re⁡(y)−im⁡(y)−1)((re⁡(y))2−(im⁡(y))2+1)2+4(re⁡(y))2(im⁡(y))2+2(−(re⁡(y))2+re⁡(y)+(im⁡(y))2+7im⁡(y)+6)re⁡(y)im⁡(y)((re⁡(y))2−(im⁡(y))2+1)2+4(re⁡(y))2(im⁡(y))2x_{1} = i \left(\frac{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right) \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 7 \operatorname{im}{\left(y\right)} + 6\right)}{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 7 \operatorname{re}{\left(y\right)} - \operatorname{im}{\left(y\right)} - 1\right) \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right) \left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 7 \operatorname{re}{\left(y\right)} - \operatorname{im}{\left(y\right)} - 1\right)}{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{2 \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 7 \operatorname{im}{\left(y\right)} + 6\right) \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} 
       //      2        2   \ /       2        2                     \                                                       \   /      2        2   \                                            /       2        2                     \            
       |\1 + re (y) - im (y)/*\-6 + re (y) - im (y) - re(y) + 7*im(y)/   2*(-1 + 7*re(y) - 2*im(y)*re(y) + im(y))*im(y)*re(y)|   \1 + re (y) - im (y)/*(-1 + 7*re(y) - 2*im(y)*re(y) + im(y))   2*\-6 + re (y) - im (y) - re(y) + 7*im(y)/*im(y)*re(y)
x2 = I*|-------------------------------------------------------------- - ----------------------------------------------------| + ------------------------------------------------------------ + ------------------------------------------------------
       |                                2                                                           2                        |                                  2                                                           2                         
       |           /      2        2   \        2      2                       /      2        2   \        2      2         |             /      2        2   \        2      2                       /      2        2   \        2      2          
       \           \1 + re (y) - im (y)/  + 4*im (y)*re (y)                    \1 + re (y) - im (y)/  + 4*im (y)*re (y)      /             \1 + re (y) - im (y)/  + 4*im (y)*re (y)                    \1 + re (y) - im (y)/  + 4*im (y)*re (y)
x2=i(((re⁡(y))2−(im⁡(y))2+1)((re⁡(y))2−re⁡(y)−(im⁡(y))2+7im⁡(y)−6)((re⁡(y))2−(im⁡(y))2+1)2+4(re⁡(y))2(im⁡(y))2−2(−2re⁡(y)im⁡(y)+7re⁡(y)+im⁡(y)−1)re⁡(y)im⁡(y)((re⁡(y))2−(im⁡(y))2+1)2+4(re⁡(y))2(im⁡(y))2)+((re⁡(y))2−(im⁡(y))2+1)(−2re⁡(y)im⁡(y)+7re⁡(y)+im⁡(y)−1)((re⁡(y))2−(im⁡(y))2+1)2+4(re⁡(y))2(im⁡(y))2+2((re⁡(y))2−re⁡(y)−(im⁡(y))2+7im⁡(y)−6)re⁡(y)im⁡(y)((re⁡(y))2−(im⁡(y))2+1)2+4(re⁡(y))2(im⁡(y))2x_{2} = i \left(\frac{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right) \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 7 \operatorname{im}{\left(y\right)} - 6\right)}{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 7 \operatorname{re}{\left(y\right)} + \operatorname{im}{\left(y\right)} - 1\right) \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right) \left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 7 \operatorname{re}{\left(y\right)} + \operatorname{im}{\left(y\right)} - 1\right)}{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{2 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 7 \operatorname{im}{\left(y\right)} - 6\right) \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} 
x2 = i*((re(y)^2 - im(y)^2 + 1)*(re(y)^2 - re(y) - im(y)^2 + 7*im(y) - 6)/((re(y)^2 - im(y)^2 + 1)^2 + 4*re(y)^2*im(y)^2) - 2*(-2*re(y)*im(y) + 7*re(y) + im(y) - 1)*re(y)*im(y)/((re(y)^2 - im(y)^2 + 1)^2 + 4*re(y)^2*im(y)^2)) + (re(y)^2 - im(y)^2 + 1)*(-2*re(y)*im(y) + 7*re(y) + im(y) - 1)/((re(y)^2 - im(y)^2 + 1)^2 + 4*re(y)^2*im(y)^2) + 2*(re(y)^2 - re(y) - im(y)^2 + 7*im(y) - 6)*re(y)*im(y)/((re(y)^2 - im(y)^2 + 1)^2 + 4*re(y)^2*im(y)^2)
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