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- La ecuación:
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Ecuación 5*x^2+2*x-7=0
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Ecuación 12,4-x=16
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Ecuación x+120=40*5
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Ecuación y-7/12=4
- Expresar {x} en función de y en la ecuación:
- -18*x-8*y=-18
- -13*x+12*y=-19
- -1*x+11*y=-9
- 16*x-11*y=13
-
Expresiones idénticas
- -y-log(y)=Const-((x^ dos)/ dos)+x
- menos y menos logaritmo de (y) es igual a Const menos ((x al cuadrado ) dividir por 2) más x
- menos y menos logaritmo de (y) es igual a Const menos ((x en el grado dos) dividir por dos) más x
- -y-log(y)=Const-((x2)/2)+x
- -y-logy=Const-x2/2+x
- -y-log(y)=Const-((x²)/2)+x
- -y-log(y)=Const-((x en el grado 2)/2)+x
- -y-logy=Const-x^2/2+x
- -y-log(y)=Const-((x^2) dividir por 2)+x
-
Expresiones semejantes
- -y+log(y)=Const-((x^2)/2)+x
- -y-log(y)=Const+((x^2)/2)+x
- -y-log(y)=Const-((x^2)/2)-x
- y-log(y)=Const-((x^2)/2)+x
-
Expresiones con funciones
- Logaritmo log
- log(2)/log(x)-log(3)/log(x)=4
- log4(x^2+x-25)=-log4(1/x)
- log_x-6(x^2-12)=2
- log3(x-1)=-2
- log(z+1)=pi*i
-y-log(y)=Const-((x^2)/2)+x la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2
x
-y - log(y) = c - -- + x
2
$$- y - \log{\left(y \right)} = x + \left(c - \frac{x^{2}}{2}\right)$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$- y - \log{\left(y \right)} = x + \left(c - \frac{x^{2}}{2}\right)$$
en
$$\left(- x + \left(- c + \frac{x^{2}}{2}\right)\right) + \left(- y - \log{\left(y \right)}\right) = 0$$
Abramos la expresión en la ecuación
$$\left(- x + \left(- c + \frac{x^{2}}{2}\right)\right) + \left(- y - \log{\left(y \right)}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- c + \frac{x^{2}}{2} - x - y - \log{\left(y \right)} = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = \frac{1}{2}$$
$$b = -1$$
$$c = - c - y - \log{\left(y \right)}$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \sqrt{2 c + 2 y + 2 \log{\left(y \right)} + 1} + 1$$
$$x_{2} = 1 - \sqrt{2 c + 2 y + 2 \log{\left(y \right)} + 1}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$- y - \log{\left(y \right)} = x + \left(c - \frac{x^{2}}{2}\right)$$
en
$$\left(- x + \left(- c + \frac{x^{2}}{2}\right)\right) + \left(- y - \log{\left(y \right)}\right) = 0$$
Abramos la expresión en la ecuación
$$\left(- x + \left(- c + \frac{x^{2}}{2}\right)\right) + \left(- y - \log{\left(y \right)}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- c + \frac{x^{2}}{2} - x - y - \log{\left(y \right)} = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = \frac{1}{2}$$
$$b = -1$$
$$c = - c - y - \log{\left(y \right)}$$
, entonces
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1/2) * (-c - y - log(y)) = 1 + 2*c + 2*y + 2*log(y)
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \sqrt{2 c + 2 y + 2 \log{\left(y \right)} + 1} + 1$$
$$x_{2} = 1 - \sqrt{2 c + 2 y + 2 \log{\left(y \right)} + 1}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$- y - \log{\left(y \right)} = x + \left(c - \frac{x^{2}}{2}\right)$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$- 2 c + x^{2} - 2 x - 2 y - 2 \log{\left(y \right)} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -2$$
$$q = \frac{c}{a}$$
$$q = - 2 c - 2 y - 2 \log{\left(y \right)}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 2$$
$$x_{1} x_{2} = - 2 c - 2 y - 2 \log{\left(y \right)}$$
$$- y - \log{\left(y \right)} = x + \left(c - \frac{x^{2}}{2}\right)$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$- 2 c + x^{2} - 2 x - 2 y - 2 \log{\left(y \right)} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -2$$
$$q = \frac{c}{a}$$
$$q = - 2 c - 2 y - 2 \log{\left(y \right)}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 2$$
$$x_{1} x_{2} = - 2 c - 2 y - 2 \log{\left(y \right)}$$
Gráfica
Respuesta rápida
[src]
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4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\ 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\
x1 = 1 - \/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *cos|-----------------------------------------------------------------------| - I*\/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *sin|-----------------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1$$
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4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\ 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\
x2 = 1 + \/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *cos|-----------------------------------------------------------------------| + I*\/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *sin|-----------------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1$$
x2 = i*((2*im(c) + 2*im(y) + 2*arg(y))^2 + (2*log(|y|) + 2*re(c) + 2*re(y) + 1)^2)^(1/4)*sin(atan2(2*im(c) + 2*im(y) + 2*arg(y, 2*log(|y|) + 2*re(c) + 2*re(y) + 1)/2) + ((2*im(c) + 2*im(y) + 2*arg(y))^2 + (2*log(|y|) + 2*re(c) + 2*re(y) + 1)^2)^(1/4)*cos(atan2(2*im(c) + 2*im(y) + 2*arg(y), 2*log(|y|) + 2*re(c) + 2*re(y) + 1)/2) + 1)
Suma y producto de raíces
[src]
suma
_________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________
4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\ 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\ 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\ 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\
1 - \/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *cos|-----------------------------------------------------------------------| - I*\/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *sin|-----------------------------------------------------------------------| + 1 + \/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *cos|-----------------------------------------------------------------------| + I*\/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *sin|-----------------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1\right) + \left(i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1\right)$$
=
2
$$2$$
producto
/ _________________________________________________________________________ _________________________________________________________________________ \ / _________________________________________________________________________ _________________________________________________________________________ \ | 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\ 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\| | 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\ 4 / 2 2 /atan2(2*arg(y) + 2*im(c) + 2*im(y), 1 + 2*log(|y|) + 2*re(c) + 2*re(y))\| |1 - \/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *cos|-----------------------------------------------------------------------| - I*\/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *sin|-----------------------------------------------------------------------||*|1 + \/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *cos|-----------------------------------------------------------------------| + I*\/ (2*arg(y) + 2*im(c) + 2*im(y)) + (1 + 2*log(|y|) + 2*re(c) + 2*re(y)) *sin|-----------------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1\right) \left(i \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)}\right)^{2} + \left(2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(c\right)} + 2 \operatorname{im}{\left(y\right)} + 2 \arg{\left(y \right)},2 \log{\left(\left|{y}\right| \right)} + 2 \operatorname{re}{\left(c\right)} + 2 \operatorname{re}{\left(y\right)} + 1 \right)}}{2} \right)} + 1\right)$$
=
-2*log(|y|) - 2*re(c) - 2*re(y) - 2*I*arg(y) - 2*I*im(c) - 2*I*im(y)
$$- 2 \log{\left(\left|{y}\right| \right)} - 2 \operatorname{re}{\left(c\right)} - 2 \operatorname{re}{\left(y\right)} - 2 i \operatorname{im}{\left(c\right)} - 2 i \operatorname{im}{\left(y\right)} - 2 i \arg{\left(y \right)}$$
-2*log(|y|) - 2*re(c) - 2*re(y) - 2*i*arg(y) - 2*i*im(c) - 2*i*im(y)