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Xy=1+x^(2) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
           2
x*y = 1 + x
xy=x2+1x y = x^{2} + 1
Simplificar
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.

La ecuación se convierte de
xy=x2+1x y = x^{2} + 1
en
xy+(−x2−1)=0x y + \left(- x^{2} - 1\right) = 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:
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=−1a = -1
b=yb = y
c=−1c = -1
, entonces
D = b^2 - 4 * a * c = 

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

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

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

o
x1=y2−y2−42x_{1} = \frac{y}{2} - \frac{\sqrt{y^{2} - 4}}{2}
x2=y2+y2−42x_{2} = \frac{y}{2} + \frac{\sqrt{y^{2} - 4}}{2}
Teorema de Cardano-Vieta
reescribamos la ecuación
xy=x2+1x y = x^{2} + 1
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
x2−xy+1=0x^{2} - x y + 1 = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=−yp = - y
q=caq = \frac{c}{a}
q=1q = 1
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=yx_{1} + x_{2} = y
x1x2=1x_{1} x_{2} = 1
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), -4 + re (y) - im (y)/||   4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|             |        4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/||   4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|
          |        \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||   \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------|             |        \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||   \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------|
re(y)     |im(y)                                                      \                    2                     /|                                                      \                    2                     /   re(y)     |im(y)                                                      \                    2                     /|                                                      \                    2                     /
----- + I*|----- - -----------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------- + ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
  2       \  2                                                    2                                               /                                                  2                                                    2       \  2                                                    2                                               /                                                  2
(i(−((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+im⁡(y)2)−((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+re⁡(y)2)+(i(((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+im⁡(y)2)+((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+re⁡(y)2)\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}\right) 
=
  /            ___________________________________________                                                \     /            ___________________________________________                                                \        
  |           /                       2                       /     /                      2        2   \\|     |           /                       2                       /     /                      2        2   \\|        
  |        4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/||     |        4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/||        
  |        \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||     |        \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||        
  |im(y)                                                      \                    2                     /|     |im(y)                                                      \                    2                     /|        
I*|----- + -----------------------------------------------------------------------------------------------| + I*|----- - -----------------------------------------------------------------------------------------------| + re(y)
  \  2                                                    2                                               /     \  2                                                    2                                               /
i(−((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+im⁡(y)2)+i(((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+im⁡(y)2)+re⁡(y)i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \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), -4 + re (y) - im (y)/||   4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|| |          |        4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/||   4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/||
|          |        \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||   \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------|| |          |        \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||   \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------||
|re(y)     |im(y)                                                      \                    2                     /|                                                      \                    2                     /| |re(y)     |im(y)                                                      \                    2                     /|                                                      \                    2                     /|
|----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------|*|----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------|
\  2       \  2                                                    2                                               /                                                  2                                               / \  2       \  2                                                    2                                               /                                                  2                                               /
(i(−((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+im⁡(y)2)−((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+re⁡(y)2)(i(((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+im⁡(y)2)+((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+re⁡(y)2)\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2}\right) 
=
1
11 
1
Respuesta rápida [src] 
               /            ___________________________________________                                                \       ___________________________________________                                                
               |           /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
               |        4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/||   4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|
               |        \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||   \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------|
     re(y)     |im(y)                                                      \                    2                     /|                                                      \                    2                     /
x1 = ----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------
       2       \  2                                                    2                                               /                                                  2
x1=i(−((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+im⁡(y)2)−((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+re⁡(y)2x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2} 
               /            ___________________________________________                                                \       ___________________________________________                                                
               |           /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
               |        4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/||   4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), -4 + re (y) - im (y)/|
               |        \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||   \/   \-4 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------|
     re(y)     |im(y)                                                      \                    2                     /|                                                      \                    2                     /
x2 = ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
       2       \  2                                                    2                                               /                                                  2
x2=i(((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+im⁡(y)2)+((re⁡(y))2−(im⁡(y))2−4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2−4)2)2+re⁡(y)2x_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 4\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} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(y\right)}}{2} 
x2 = i*(((re(y)^2 - im(y)^2 - 4)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(2*re(y)*im(y, re(y)^2 - im(y)^2 - 4)/2)/2 + im(y)/2) + ((re(y)^2 - im(y)^2 - 4)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(2*re(y)*im(y), re(y)^2 - im(y)^2 - 4)/2)/2 + re(y)/2)
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