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x^2-1=1+(1/2)*y la ecuación

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Solución numérica:

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Solución

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

(0)^2 - 4 * (1) * (-2 - y/2) = 8 + 2*y

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

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

o
x1=2y+82x_{1} = \frac{\sqrt{2 y + 8}}{2}
x2=−2y+82x_{2} = - \frac{\sqrt{2 y + 8}}{2}
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=−y2−2q = - \frac{y}{2} - 2
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=0x_{1} + x_{2} = 0
x1x2=−y2−2x_{1} x_{2} = - \frac{y}{2} - 2
Gráfica
Respuesta rápida [src] 
          ___________________________                                         ___________________________                                 
       4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\     4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\
       \/  (8 + 2*re(y))  + 4*im (y) *cos|---------------------------|   I*\/  (8 + 2*re(y))  + 4*im (y) *sin|---------------------------|
                                         \             2             /                                       \             2             /
x1 = - --------------------------------------------------------------- - -----------------------------------------------------------------
                                      2                                                                  2
x1=−i(2re⁡(y)+8)2+4(im⁡(y))24sin⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2−(2re⁡(y)+8)2+4(im⁡(y))24cos⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2x_{1} = - \frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} 
        ___________________________                                         ___________________________                                 
     4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\     4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\
     \/  (8 + 2*re(y))  + 4*im (y) *cos|---------------------------|   I*\/  (8 + 2*re(y))  + 4*im (y) *sin|---------------------------|
                                       \             2             /                                       \             2             /
x2 = --------------------------------------------------------------- + -----------------------------------------------------------------
                                    2                                                                  2
x2=i(2re⁡(y)+8)2+4(im⁡(y))24sin⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2+(2re⁡(y)+8)2+4(im⁡(y))24cos⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2x_{2} = \frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} 
x2 = i*((2*re(y) + 8)^2 + 4*im(y)^2)^(1/4)*sin(atan2(2*im(y, 2*re(y) + 8)/2)/2 + ((2*re(y) + 8)^2 + 4*im(y)^2)^(1/4)*cos(atan2(2*im(y), 2*re(y) + 8)/2)/2)
Suma y producto de raíces [src] 
suma
     ___________________________                                         ___________________________                                       ___________________________                                         ___________________________                                 
  4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\     4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\   4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\     4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\
  \/  (8 + 2*re(y))  + 4*im (y) *cos|---------------------------|   I*\/  (8 + 2*re(y))  + 4*im (y) *sin|---------------------------|   \/  (8 + 2*re(y))  + 4*im (y) *cos|---------------------------|   I*\/  (8 + 2*re(y))  + 4*im (y) *sin|---------------------------|
                                    \             2             /                                       \             2             /                                     \             2             /                                       \             2             /
- --------------------------------------------------------------- - ----------------------------------------------------------------- + --------------------------------------------------------------- + -----------------------------------------------------------------
                                 2                                                                  2                                                                  2                                                                  2
(−i(2re⁡(y)+8)2+4(im⁡(y))24sin⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2−(2re⁡(y)+8)2+4(im⁡(y))24cos⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2)+(i(2re⁡(y)+8)2+4(im⁡(y))24sin⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2+(2re⁡(y)+8)2+4(im⁡(y))24cos⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2)\left(- \frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}\right) + \left(\frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}\right) 
=
0
00 
producto
/     ___________________________                                         ___________________________                                 \ /   ___________________________                                         ___________________________                                 \
|  4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\     4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\| |4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\     4 /              2       2        /atan2(2*im(y), 8 + 2*re(y))\|
|  \/  (8 + 2*re(y))  + 4*im (y) *cos|---------------------------|   I*\/  (8 + 2*re(y))  + 4*im (y) *sin|---------------------------|| |\/  (8 + 2*re(y))  + 4*im (y) *cos|---------------------------|   I*\/  (8 + 2*re(y))  + 4*im (y) *sin|---------------------------||
|                                    \             2             /                                       \             2             /| |                                  \             2             /                                       \             2             /|
|- --------------------------------------------------------------- - -----------------------------------------------------------------|*|--------------------------------------------------------------- + -----------------------------------------------------------------|
\                                 2                                                                  2                                / \                               2                                                                  2                                /
(−i(2re⁡(y)+8)2+4(im⁡(y))24sin⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2−(2re⁡(y)+8)2+4(im⁡(y))24cos⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2)(i(2re⁡(y)+8)2+4(im⁡(y))24sin⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2+(2re⁡(y)+8)2+4(im⁡(y))24cos⁡(atan2⁡(2im⁡(y),2re⁡(y)+8)2)2)\left(- \frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}\right) \left(\frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}\right) 
=
    _______________________                                
   /            2     2      I*atan2(2*im(y), 8 + 2*re(y)) 
-\/  (4 + re(y))  + im (y) *e                              
-----------------------------------------------------------
                             2
−(re⁡(y)+4)2+(im⁡(y))2eiatan2⁡(2im⁡(y),2re⁡(y)+8)2- \frac{\sqrt{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}}{2} 
-sqrt((4 + re(y))^2 + im(y)^2)*exp(i*atan2(2*im(y), 8 + 2*re(y)))/2
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