Sr Examen

Otras calculadoras

(-3)*y^2/2+6*y=c-log(x^2+3)/2 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src] 
    2                / 2    \
-3*y              log\x  + 3/
----- + 6*y = c - -----------
  2                    2
6y+(−1)3y22=c−log⁡(x2+3)26 y + \frac{\left(-1\right) 3 y^{2}}{2} = c - \frac{\log{\left(x^{2} + 3 \right)}}{2}
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
6y+(−1)3y22=c−log⁡(x2+3)26 y + \frac{\left(-1\right) 3 y^{2}}{2} = c - \frac{\log{\left(x^{2} + 3 \right)}}{2}
en
(−c+log⁡(x2+3)2)+(6y+(−1)3y22)=0\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
(−c+log⁡(x2+3)2)+(6y+(−1)3y22)=0\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−3y22+6y+log⁡(x2+3)2=0- 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:
y1=D−b2ay_{1} = \frac{\sqrt{D} - b}{2 a}
y2=−D−b2ay_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−32a = - \frac{3}{2}
b=6b = 6
c=−c+log⁡(x2+3)2c = - 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
y1=2−−6c+3log⁡(x2+3)+363y_{1} = 2 - \frac{\sqrt{- 6 c + 3 \log{\left(x^{2} + 3 \right)} + 36}}{3}
y2=−6c+3log⁡(x2+3)+363+2y_{2} = \frac{\sqrt{- 6 c + 3 \log{\left(x^{2} + 3 \right)} + 36}}{3} + 2
Teorema de Cardano-Vieta
reescribamos la ecuación
6y+(−1)3y22=c−log⁡(x2+3)26 y + \frac{\left(-1\right) 3 y^{2}}{2} = c - \frac{\log{\left(x^{2} + 3 \right)}}{2}
de
ay2+by+c=0a y^{2} + b y + c = 0
como ecuación cuadrática reducida
y2+bya+ca=0y^{2} + \frac{b y}{a} + \frac{c}{a} = 0
2c3+y2−4y−log⁡(x2+3)3=0\frac{2 c}{3} + y^{2} - 4 y - \frac{\log{\left(x^{2} + 3 \right)}}{3} = 0
py+q+y2=0p y + q + y^{2} = 0
donde
p=bap = \frac{b}{a}
p=−4p = -4
q=caq = \frac{c}{a}
q=2c3−log⁡(x2+3)3q = \frac{2 c}{3} - \frac{\log{\left(x^{2} + 3 \right)}}{3}
Fórmulas de Cardano-Vieta
y1+y2=−py_{1} + y_{2} = - p
y1y2=qy_{1} y_{2} = q
y1+y2=4y_{1} + y_{2} = 4
y1y2=2c3−log⁡(x2+3)3y_{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
(−i(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24sin⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3−(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24cos⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+2)+(i(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24sin⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24cos⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+2)\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
44 
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                                                                      /
(−i(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24sin⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3−(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24cos⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+2)(i(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24sin⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24cos⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+2)\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
−log⁡(∣x2+3∣)3+2re⁡(c)3+2iim⁡(c)3−iarg⁡(x2+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
y1=−i(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24sin⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3−(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24cos⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+2y_{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
y2=i(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24sin⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+(−6im⁡(c)+3arg⁡(x2+3))2+(3log⁡(∣x2+3∣)−6re⁡(c)+36)24cos⁡(atan2⁡(−6im⁡(c)+3arg⁡(x2+3),3log⁡(∣x2+3∣)−6re⁡(c)+36)2)3+2y_{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)
    © Sr Examen – Калькулятор