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log(y)=((x^2)/2)+x la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
          2    
         x     
log(y) = -- + x
         2
log⁡(y)=x22+x\log{\left(y \right)} = \frac{x^{2}}{2} + x
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
log⁡(y)=x22+x\log{\left(y \right)} = \frac{x^{2}}{2} + x
en
(−x22−x)+log⁡(y)=0\left(- \frac{x^{2}}{2} - x\right) + \log{\left(y \right)} = 0
Abramos la expresión en la ecuación
(−x22−x)+log⁡(y)=0\left(- \frac{x^{2}}{2} - x\right) + \log{\left(y \right)} = 0
Obtenemos la ecuación cuadrática
−x22−x+log⁡(y)=0- \frac{x^{2}}{2} - x + \log{\left(y \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=−12a = - \frac{1}{2}
b=−1b = -1
c=log⁡(y)c = \log{\left(y \right)}
, entonces
D = b^2 - 4 * a * c = 

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

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

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

o
x1=−2log⁡(y)+1−1x_{1} = - \sqrt{2 \log{\left(y \right)} + 1} - 1
x2=2log⁡(y)+1−1x_{2} = \sqrt{2 \log{\left(y \right)} + 1} - 1
Teorema de Cardano-Vieta
reescribamos la ecuación
log⁡(y)=x22+x\log{\left(y \right)} = \frac{x^{2}}{2} + x
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+2x−2log⁡(y)=0x^{2} + 2 x - 2 \log{\left(y \right)} = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=2p = 2
q=caq = \frac{c}{a}
q=−2log⁡(y)q = - 2 \log{\left(y \right)}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−2x_{1} + x_{2} = -2
x1x2=−2log⁡(y)x_{1} x_{2} = - 2 \log{\left(y \right)}
Gráfica
Respuesta rápida [src] 
             _______________________________                                             _______________________________                                     
          4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\
x1 = -1 - \/  (1 + 2*log(|y|))  + 4*arg (y) *cos|-------------------------------| - I*\/  (1 + 2*log(|y|))  + 4*arg (y) *sin|-------------------------------|
                                                \               2               /                                           \               2               /
x1=−i(2log⁡(∣y∣)+1)2+4arg⁡2(y)4sin⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−(2log⁡(∣y∣)+1)2+4arg⁡2(y)4cos⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−1x_{1} = - i \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - 1 
             _______________________________                                             _______________________________                                     
          4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\
x2 = -1 + \/  (1 + 2*log(|y|))  + 4*arg (y) *cos|-------------------------------| + I*\/  (1 + 2*log(|y|))  + 4*arg (y) *sin|-------------------------------|
                                                \               2               /                                           \               2               /
x2=i(2log⁡(∣y∣)+1)2+4arg⁡2(y)4sin⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)+(2log⁡(∣y∣)+1)2+4arg⁡2(y)4cos⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−1x_{2} = i \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - 1 
x2 = i*((2*log(|y|) + 1)^2 + 4*arg(y)^2)^(1/4)*sin(atan2(2*arg(y, 2*log(|y|) + 1)/2) + ((2*log(|y|) + 1)^2 + 4*arg(y)^2)^(1/4)*cos(atan2(2*arg(y), 2*log(|y|) + 1)/2) - 1)
Suma y producto de raíces [src] 
suma
        _______________________________                                             _______________________________                                                _______________________________                                             _______________________________                                     
     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\        4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\
-1 - \/  (1 + 2*log(|y|))  + 4*arg (y) *cos|-------------------------------| - I*\/  (1 + 2*log(|y|))  + 4*arg (y) *sin|-------------------------------| + -1 + \/  (1 + 2*log(|y|))  + 4*arg (y) *cos|-------------------------------| + I*\/  (1 + 2*log(|y|))  + 4*arg (y) *sin|-------------------------------|
                                           \               2               /                                           \               2               /                                              \               2               /                                           \               2               /
(−i(2log⁡(∣y∣)+1)2+4arg⁡2(y)4sin⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−(2log⁡(∣y∣)+1)2+4arg⁡2(y)4cos⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−1)+(i(2log⁡(∣y∣)+1)2+4arg⁡2(y)4sin⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)+(2log⁡(∣y∣)+1)2+4arg⁡2(y)4cos⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−1)\left(- i \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - 1\right) + \left(i \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - 1\right) 
=
-2
−2-2 
producto
/        _______________________________                                             _______________________________                                     \ /        _______________________________                                             _______________________________                                     \
|     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\| |     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\     4 /                 2        2        /atan2(2*arg(y), 1 + 2*log(|y|))\|
|-1 - \/  (1 + 2*log(|y|))  + 4*arg (y) *cos|-------------------------------| - I*\/  (1 + 2*log(|y|))  + 4*arg (y) *sin|-------------------------------||*|-1 + \/  (1 + 2*log(|y|))  + 4*arg (y) *cos|-------------------------------| + I*\/  (1 + 2*log(|y|))  + 4*arg (y) *sin|-------------------------------||
\                                           \               2               /                                           \               2               // \                                           \               2               /                                           \               2               //
(−i(2log⁡(∣y∣)+1)2+4arg⁡2(y)4sin⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−(2log⁡(∣y∣)+1)2+4arg⁡2(y)4cos⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−1)(i(2log⁡(∣y∣)+1)2+4arg⁡2(y)4sin⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)+(2log⁡(∣y∣)+1)2+4arg⁡2(y)4cos⁡(atan2⁡(2arg⁡(y),2log⁡(∣y∣)+1)2)−1)\left(- i \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - 1\right) \left(i \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \log{\left(\left|{y}\right| \right)} + 1\right)^{2} + 4 \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 1 \right)}}{2} \right)} - 1\right) 
=
-2*log(|y|) - 2*I*arg(y)
−2log⁡(∣y∣)−2iarg⁡(y)- 2 \log{\left(\left|{y}\right| \right)} - 2 i \arg{\left(y \right)} 
-2*log(|y|) - 2*i*arg(y)
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