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log(y)=-x^2/4 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
           2 
         -x  
log(y) = ----
          4
log⁡(y)=(−1)x24\log{\left(y \right)} = \frac{\left(-1\right) x^{2}}{4}
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)=(−1)x24\log{\left(y \right)} = \frac{\left(-1\right) x^{2}}{4}
en
−(−1)x24+log⁡(y)=0- \frac{\left(-1\right) x^{2}}{4} + \log{\left(y \right)} = 0
Abramos la expresión en la ecuación
−(−1)x24+log⁡(y)=0- \frac{\left(-1\right) x^{2}}{4} + \log{\left(y \right)} = 0
Obtenemos la ecuación cuadrática
x24+log⁡(y)=0\frac{x^{2}}{4} + \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=14a = \frac{1}{4}
b=0b = 0
c=log⁡(y)c = \log{\left(y \right)}
, entonces
D = b^2 - 4 * a * c = 

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

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

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

o
x1=2−log⁡(y)x_{1} = 2 \sqrt{- \log{\left(y \right)}}
x2=−2−log⁡(y)x_{2} = - 2 \sqrt{- \log{\left(y \right)}}
Teorema de Cardano-Vieta
reescribamos la ecuación
log⁡(y)=(−1)x24\log{\left(y \right)} = \frac{\left(-1\right) x^{2}}{4}
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+4log⁡(y)=0x^{2} + 4 \log{\left(y \right)} = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=4log⁡(y)q = 4 \log{\left(y \right)}
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=4log⁡(y)x_{1} x_{2} = 4 \log{\left(y \right)}
Gráfica
Suma y producto de raíces [src] 
suma
       _____________________                                         _____________________                                       _____________________                                         _____________________                               
    4 /    2         2          /atan2(-arg(y), -log(|y|))\       4 /    2         2          /atan2(-arg(y), -log(|y|))\     4 /    2         2          /atan2(-arg(y), -log(|y|))\       4 /    2         2          /atan2(-arg(y), -log(|y|))\
- 2*\/  arg (y) + log (|y|) *cos|-------------------------| - 2*I*\/  arg (y) + log (|y|) *sin|-------------------------| + 2*\/  arg (y) + log (|y|) *cos|-------------------------| + 2*I*\/  arg (y) + log (|y|) *sin|-------------------------|
                                \            2            /                                   \            2            /                                 \            2            /                                   \            2            /
(−2ilog⁡(∣y∣)2+arg⁡2(y)4sin⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2)−2log⁡(∣y∣)2+arg⁡2(y)4cos⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2))+(2ilog⁡(∣y∣)2+arg⁡2(y)4sin⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2)+2log⁡(∣y∣)2+arg⁡2(y)4cos⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2))\left(- 2 i \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)} - 2 \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)}\right) + \left(2 i \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)} + 2 \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)}\right) 
=
0
00 
producto
/       _____________________                                         _____________________                               \ /     _____________________                                         _____________________                               \
|    4 /    2         2          /atan2(-arg(y), -log(|y|))\       4 /    2         2          /atan2(-arg(y), -log(|y|))\| |  4 /    2         2          /atan2(-arg(y), -log(|y|))\       4 /    2         2          /atan2(-arg(y), -log(|y|))\|
|- 2*\/  arg (y) + log (|y|) *cos|-------------------------| - 2*I*\/  arg (y) + log (|y|) *sin|-------------------------||*|2*\/  arg (y) + log (|y|) *cos|-------------------------| + 2*I*\/  arg (y) + log (|y|) *sin|-------------------------||
\                                \            2            /                                   \            2            // \                              \            2            /                                   \            2            //
(−2ilog⁡(∣y∣)2+arg⁡2(y)4sin⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2)−2log⁡(∣y∣)2+arg⁡2(y)4cos⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2))(2ilog⁡(∣y∣)2+arg⁡2(y)4sin⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2)+2log⁡(∣y∣)2+arg⁡2(y)4cos⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2))\left(- 2 i \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)} - 2 \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)}\right) \left(2 i \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)} + 2 \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)}\right) 
=
      _____________________                             
     /    2         2        I*atan2(-arg(y), -log(|y|))
-4*\/  arg (y) + log (|y|) *e
−4log⁡(∣y∣)2+arg⁡2(y)eiatan2⁡(−arg⁡(y),−log⁡(∣y∣))- 4 \sqrt{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} e^{i \operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}} 
-4*sqrt(arg(y)^2 + log(|y|)^2)*exp(i*atan2(-arg(y), -log(|y|)))
Respuesta rápida [src] 
            _____________________                                         _____________________                               
         4 /    2         2          /atan2(-arg(y), -log(|y|))\       4 /    2         2          /atan2(-arg(y), -log(|y|))\
x1 = - 2*\/  arg (y) + log (|y|) *cos|-------------------------| - 2*I*\/  arg (y) + log (|y|) *sin|-------------------------|
                                     \            2            /                                   \            2            /
x1=−2ilog⁡(∣y∣)2+arg⁡2(y)4sin⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2)−2log⁡(∣y∣)2+arg⁡2(y)4cos⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2)x_{1} = - 2 i \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)} - 2 \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)} 
          _____________________                                         _____________________                               
       4 /    2         2          /atan2(-arg(y), -log(|y|))\       4 /    2         2          /atan2(-arg(y), -log(|y|))\
x2 = 2*\/  arg (y) + log (|y|) *cos|-------------------------| + 2*I*\/  arg (y) + log (|y|) *sin|-------------------------|
                                   \            2            /                                   \            2            /
x2=2ilog⁡(∣y∣)2+arg⁡2(y)4sin⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2)+2log⁡(∣y∣)2+arg⁡2(y)4cos⁡(atan2⁡(−arg⁡(y),−log⁡(∣y∣))2)x_{2} = 2 i \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)} + 2 \sqrt[4]{\log{\left(\left|{y}\right| \right)}^{2} + \arg^{2}{\left(y \right)}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \arg{\left(y \right)},- \log{\left(\left|{y}\right| \right)} \right)}}{2} \right)} 
x2 = 2*i*(log(|y|)^2 + arg(y)^2)^(1/4)*sin(atan2(-arg(y, -log(|y|))/2) + 2*(log(|y|)^2 + arg(y)^2)^(1/4)*cos(atan2(-arg(y), -log(|y|))/2))
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