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(-3)*y^2/2+6*y=c-log(x^2+3)/2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 / 2 \ -3*y log\x + 3/ ----- + 6*y = c - ----------- 2 2
$$6 y + \frac{\left(-1\right) 3 y^{2}}{2} = c - \frac{\log{\left(x^{2} + 3 \right)}}{2}$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$6 y + \frac{\left(-1\right) 3 y^{2}}{2} = c - \frac{\log{\left(x^{2} + 3 \right)}}{2}$$
en
$$\left(- c + \frac{\log{\left(x^{2} + 3 \right)}}{2}\right) + \left(6 y + \frac{\left(-1\right) 3 y^{2}}{2}\right) = 0$$
Abramos la expresión en la ecuación
$$\left(- c + \frac{\log{\left(x^{2} + 3 \right)}}{2}\right) + \left(6 y + \frac{\left(-1\right) 3 y^{2}}{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- c - \frac{3 y^{2}}{2} + 6 y + \frac{\log{\left(x^{2} + 3 \right)}}{2} = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = - \frac{3}{2}$$
$$b = 6$$
$$c = - c + \frac{\log{\left(x^{2} + 3 \right)}}{2}$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = 2 - \frac{\sqrt{- 6 c + 3 \log{\left(x^{2} + 3 \right)} + 36}}{3}$$
$$y_{2} = \frac{\sqrt{- 6 c + 3 \log{\left(x^{2} + 3 \right)} + 36}}{3} + 2$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$6 y + \frac{\left(-1\right) 3 y^{2}}{2} = c - \frac{\log{\left(x^{2} + 3 \right)}}{2}$$
en
$$\left(- c + \frac{\log{\left(x^{2} + 3 \right)}}{2}\right) + \left(6 y + \frac{\left(-1\right) 3 y^{2}}{2}\right) = 0$$
Abramos la expresión en la ecuación
$$\left(- c + \frac{\log{\left(x^{2} + 3 \right)}}{2}\right) + \left(6 y + \frac{\left(-1\right) 3 y^{2}}{2}\right) = 0$$
Obtenemos la ecuación cuadrática
$$- c - \frac{3 y^{2}}{2} + 6 y + \frac{\log{\left(x^{2} + 3 \right)}}{2} = 0$$
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:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = - \frac{3}{2}$$
$$b = 6$$
$$c = - c + \frac{\log{\left(x^{2} + 3 \right)}}{2}$$
, entonces
D = b^2 - 4 * a * c =
(6)^2 - 4 * (-3/2) * (log(3 + x^2)/2 - c) = 36 - 6*c + 3*log(3 + x^2)
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = 2 - \frac{\sqrt{- 6 c + 3 \log{\left(x^{2} + 3 \right)} + 36}}{3}$$
$$y_{2} = \frac{\sqrt{- 6 c + 3 \log{\left(x^{2} + 3 \right)} + 36}}{3} + 2$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$6 y + \frac{\left(-1\right) 3 y^{2}}{2} = c - \frac{\log{\left(x^{2} + 3 \right)}}{2}$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$\frac{2 c}{3} + y^{2} - 4 y - \frac{\log{\left(x^{2} + 3 \right)}}{3} = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = \frac{2 c}{3} - \frac{\log{\left(x^{2} + 3 \right)}}{3}$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = 4$$
$$y_{1} y_{2} = \frac{2 c}{3} - \frac{\log{\left(x^{2} + 3 \right)}}{3}$$
$$6 y + \frac{\left(-1\right) 3 y^{2}}{2} = c - \frac{\log{\left(x^{2} + 3 \right)}}{2}$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$\frac{2 c}{3} + y^{2} - 4 y - \frac{\log{\left(x^{2} + 3 \right)}}{3} = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = \frac{2 c}{3} - \frac{\log{\left(x^{2} + 3 \right)}}{3}$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = 4$$
$$y_{1} y_{2} = \frac{2 c}{3} - \frac{\log{\left(x^{2} + 3 \right)}}{3}$$
Gráfica
Suma y producto de raíces
[src]
suma
_________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________
/ 2 2 / / / 2\ /| 2|\\\ / 2 2 / / / 2\ /| 2|\\\ / 2 2 / / / 2\ /| 2|\\\ / 2 2 / / / 2\ /| 2|\\\
4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//| 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//| 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//| 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//|
\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *cos|---------------------------------------------------------------| I*\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *sin|---------------------------------------------------------------| \/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *cos|---------------------------------------------------------------| I*\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *sin|---------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
2 - ------------------------------------------------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------- + 2 + ------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------
3 3 3 3
$$\left(- \frac{i \sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + 2\right) + \left(\frac{i \sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + 2\right)$$
=
4
$$4$$
producto
/ _________________________________________________________________ _________________________________________________________________ \ / _________________________________________________________________ _________________________________________________________________ \ | / 2 2 / / / 2\ /| 2|\\\ / 2 2 / / / 2\ /| 2|\\\| | / 2 2 / / / 2\ /| 2|\\\ / 2 2 / / / 2\ /| 2|\\\| | 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//| 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//|| | 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//| 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//|| | \/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *cos|---------------------------------------------------------------| I*\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *sin|---------------------------------------------------------------|| | \/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *cos|---------------------------------------------------------------| I*\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *sin|---------------------------------------------------------------|| | \ 2 / \ 2 /| | \ 2 / \ 2 /| |2 - ------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------|*|2 + ------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------| \ 3 3 / \ 3 3 /
$$\left(- \frac{i \sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + 2\right) \left(\frac{i \sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + 2\right)$$
=
/| 2|\ / 2\
log\|3 + x |/ 2*re(c) I*arg\3 + x / 2*I*im(c)
- ------------- + ------- - ------------- + ---------
3 3 3 3
$$- \frac{\log{\left(\left|{x^{2} + 3}\right| \right)}}{3} + \frac{2 \operatorname{re}{\left(c\right)}}{3} + \frac{2 i \operatorname{im}{\left(c\right)}}{3} - \frac{i \arg{\left(x^{2} + 3 \right)}}{3}$$
-log(|3 + x^2|)/3 + 2*re(c)/3 - i*arg(3 + x^2)/3 + 2*i*im(c)/3
Respuesta rápida
[src]
_________________________________________________________________ _________________________________________________________________
/ 2 2 / / / 2\ /| 2|\\\ / 2 2 / / / 2\ /| 2|\\\
4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//| 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//|
\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *cos|---------------------------------------------------------------| I*\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *sin|---------------------------------------------------------------|
\ 2 / \ 2 /
y1 = 2 - ------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------
3 3
$$y_{1} = - \frac{i \sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + 2$$
_________________________________________________________________ _________________________________________________________________
/ 2 2 / / / 2\ /| 2|\\\ / 2 2 / / / 2\ /| 2|\\\
4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//| 4 / / / 2\\ / /| 2|\\ |atan2\-6*im(c) + 3*arg\3 + x /, 36 - 6*re(c) + 3*log\|3 + x |//|
\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *cos|---------------------------------------------------------------| I*\/ \-6*im(c) + 3*arg\3 + x // + \36 - 6*re(c) + 3*log\|3 + x |// *sin|---------------------------------------------------------------|
\ 2 / \ 2 /
y2 = 2 + ------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------
3 3
$$y_{2} = \frac{i \sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)}\right)^{2} + \left(3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 6 \operatorname{im}{\left(c\right)} + 3 \arg{\left(x^{2} + 3 \right)},3 \log{\left(\left|{x^{2} + 3}\right| \right)} - 6 \operatorname{re}{\left(c\right)} + 36 \right)}}{2} \right)}}{3} + 2$$
y2 = i*((-6*im(c) + 3*arg(x^2 + 3))^2 + (3*log(|x^2 + 3|) - 6*re(c) + 36)^2)^(1/4)*sin(atan2(-6*im(c) + 3*arg(x^2 + 3, 3*log(|x^2 + 3|) - 6*re(c) + 36)/2)/3 + ((-6*im(c) + 3*arg(x^2 + 3))^2 + (3*log(|x^2 + 3|) - 6*re(c) + 36)^2)^(1/4)*cos(atan2(-6*im(c) + 3*arg(x^2 + 3), 3*log(|x^2 + 3|) - 6*re(c) + 36)/2)/3 + 2)