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xy=y^2+1 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
       2    
x*y = y  + 1
xy=y2+1x y = y^{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=y2+1x y = y^{2} + 1
en
xy+(−y2−1)=0x y + \left(- y^{2} - 1\right) = 0
Es la ecuación de la forma
a*y^2 + b*y + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
y1=D−b2ay_{1} = \frac{\sqrt{D} - b}{2 a}
y2=−D−b2ay_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−1a = -1
b=xb = x
c=−1c = -1
, entonces
D = b^2 - 4 * a * c = 

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

La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)

y2 = (-b - sqrt(D)) / (2*a)

o
y1=x2−x2−42y_{1} = \frac{x}{2} - \frac{\sqrt{x^{2} - 4}}{2}
y2=x2+x2−42y_{2} = \frac{x}{2} + \frac{\sqrt{x^{2} - 4}}{2}
Teorema de Cardano-Vieta
reescribamos la ecuación
xy=y2+1x y = y^{2} + 1
de
ay2+by+c=0a y^{2} + b y + c = 0
como ecuación cuadrática reducida
y2+bya+ca=0y^{2} + \frac{b y}{a} + \frac{c}{a} = 0
−xy+y2+1=0- x y + y^{2} + 1 = 0
py+q+y2=0p y + q + y^{2} = 0
donde
p=bap = \frac{b}{a}
p=−xp = - x
q=caq = \frac{c}{a}
q=1q = 1
Fórmulas de Cardano-Vieta
y1+y2=−py_{1} + y_{2} = - p
y1y2=qy_{1} y_{2} = q
y1+y2=xy_{1} + y_{2} = x
y1y2=1y_{1} y_{2} = 1
Gráfica
Respuesta rápida [src] 
               /            ___________________________________________                                                \       ___________________________________________                                                
               |           /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
               |        4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/||   4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|
               |        \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *sin|------------------------------------------||   \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *cos|------------------------------------------|
     re(x)     |im(x)                                                      \                    2                     /|                                                      \                    2                     /
y1 = ----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------
       2       \  2                                                    2                                               /                                                  2
y1=i(−((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24sin⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+im⁡(x)2)−((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24cos⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+re⁡(x)2y_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2} 
               /            ___________________________________________                                                \       ___________________________________________                                                
               |           /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
               |        4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/||   4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|
               |        \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *sin|------------------------------------------||   \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *cos|------------------------------------------|
     re(x)     |im(x)                                                      \                    2                     /|                                                      \                    2                     /
y2 = ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
       2       \  2                                                    2                                               /                                                  2
y2=i(((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24sin⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+im⁡(x)2)+((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24cos⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+re⁡(x)2y_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2} 
y2 = i*(((re(x)^2 - im(x)^2 - 4)^2 + 4*re(x)^2*im(x)^2)^(1/4)*sin(atan2(2*re(x)*im(x, re(x)^2 - im(x)^2 - 4)/2)/2 + im(x)/2) + ((re(x)^2 - im(x)^2 - 4)^2 + 4*re(x)^2*im(x)^2)^(1/4)*cos(atan2(2*re(x)*im(x), re(x)^2 - im(x)^2 - 4)/2)/2 + re(x)/2)
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(x)*re(x), -4 + re (x) - im (x)/||   4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|             |        4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/||   4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|
          |        \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *sin|------------------------------------------||   \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *cos|------------------------------------------|             |        \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *sin|------------------------------------------||   \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *cos|------------------------------------------|
re(x)     |im(x)                                                      \                    2                     /|                                                      \                    2                     /   re(x)     |im(x)                                                      \                    2                     /|                                                      \                    2                     /
----- + I*|----- - -----------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------- + ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
  2       \  2                                                    2                                               /                                                  2                                                    2       \  2                                                    2                                               /                                                  2
(i(−((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24sin⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+im⁡(x)2)−((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24cos⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+re⁡(x)2)+(i(((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24sin⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+im⁡(x)2)+((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24cos⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+re⁡(x)2)\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}\right) 
=
  /            ___________________________________________                                                \     /            ___________________________________________                                                \        
  |           /                       2                       /     /                      2        2   \\|     |           /                       2                       /     /                      2        2   \\|        
  |        4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/||     |        4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/||        
  |        \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *sin|------------------------------------------||     |        \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *sin|------------------------------------------||        
  |im(x)                                                      \                    2                     /|     |im(x)                                                      \                    2                     /|        
I*|----- + -----------------------------------------------------------------------------------------------| + I*|----- - -----------------------------------------------------------------------------------------------| + re(x)
  \  2                                                    2                                               /     \  2                                                    2                                               /
i(−((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24sin⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+im⁡(x)2)+i(((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24sin⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+im⁡(x)2)+re⁡(x)i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \operatorname{re}{\left(x\right)} 
producto
/          /            ___________________________________________                                                \       ___________________________________________                                                \ /          /            ___________________________________________                                                \       ___________________________________________                                                \
|          |           /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\| |          |           /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\|
|          |        4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/||   4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/|| |          |        4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/||   4 /  /       2        2   \        2      2        |atan2\2*im(x)*re(x), -4 + re (x) - im (x)/||
|          |        \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *sin|------------------------------------------||   \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *cos|------------------------------------------|| |          |        \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *sin|------------------------------------------||   \/   \-4 + re (x) - im (x)/  + 4*im (x)*re (x) *cos|------------------------------------------||
|re(x)     |im(x)                                                      \                    2                     /|                                                      \                    2                     /| |re(x)     |im(x)                                                      \                    2                     /|                                                      \                    2                     /|
|----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------|*|----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------|
\  2       \  2                                                    2                                               /                                                  2                                               / \  2       \  2                                                    2                                               /                                                  2                                               /
(i(−((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24sin⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+im⁡(x)2)−((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24cos⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+re⁡(x)2)(i(((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24sin⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+im⁡(x)2)+((re⁡(x))2−(im⁡(x))2−4)2+4(re⁡(x))2(im⁡(x))24cos⁡(atan2⁡(2re⁡(x)im⁡(x),(re⁡(x))2−(im⁡(x))2−4)2)2+re⁡(x)2)\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2}\right) 
=
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