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La ecuación:
Ecuación 6x+1=-4x
Ecuación x+y-3=0
Ecuación x^2=-3
Ecuación 9-7(x+3)=5-6x
Expresar {x} en función de y en la ecuación:
-11*x+9*y=-20
5*x-14*y=-15
-10*x+15*y=10
-9*x-6*y=12
Expresiones idénticas
nueve * ochenta y uno ^cosx+ tres * nueve ^cosx= cero
9 multiplicar por 81 en el grado coseno de x más 3 multiplicar por 9 en el grado coseno de x es igual a 0
nueve multiplicar por ochenta y uno en el grado coseno de x más tres multiplicar por nueve en el grado coseno de x es igual a cero
9*81cosx+3*9cosx=0
981^cosx+39^cosx=0
981cosx+39cosx=0
9*81^cosx+3*9^cosx=O
Expresiones semejantes
9*81^cosx-3*9^cosx=0
Expresiones con funciones
cosx
cosx-2|cosx|=y
cosx
cosx-2|cosx|=y

981^cosx+39^cosx=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

    cos(x)      cos(x)    
9*81       + 3*9       = 0

9 \cdot 81^{\cos{\left(x \right)}} + 3 \cdot 9^{\cos{\left(x \right)}} = 0

Simplificar

Solución detallada

Tenemos la ecuación

9 \cdot 81^{\cos{\left(x \right)}} + 3 \cdot 9^{\cos{\left(x \right)}} = 0

cambiamos

3^{2 \cos{\left(x \right)} + 1} + 3^{4 \cos{\left(x \right)} + 2} - 1 = 0

9 \cdot 81^{\cos{\left(x \right)}} + 3 \cdot 9^{\cos{\left(x \right)}} - 1 = 0

Sustituimos

w = \cos{\left(x \right)}

9 \cdot 81^{w} + 3 \cdot 9^{w} - 1 = 0

9 \cdot 81^{w} + 3 \cdot 9^{w} - 1 = 0

Sustituimos

v = 9^{w}

obtendremos

9 v^{2} + 3 v - 1 = 0

9 v^{2} + 3 v - 1 = 0

Es la ecuación de la forma

a*v^2 + b*v + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:

v_{1} = \frac{\sqrt{D} - b}{2 a}

v_{2} = \frac{- \sqrt{D} - b}{2 a}

donde D = b^2 - 4*a*c es el discriminante.
Como

a = 9

b = 3

c = -1

, entonces

D = b^2 - 4 * a * c =

(3)^2 - 4 * (9) * (-1) = 45

Como D > 0 la ecuación tiene dos raíces.

v1 = (-b + sqrt(D)) / (2*a)

v2 = (-b - sqrt(D)) / (2*a)

v_{1} = - \frac{1}{6} + \frac{\sqrt{5}}{6}

v_{2} = - \frac{\sqrt{5}}{6} - \frac{1}{6}

hacemos cambio inverso

9^{w} = v

w = \frac{\log{\left(v \right)}}{\log{\left(9 \right)}}

Entonces la respuesta definitiva es

w_{1} = \frac{\log{\left(- \frac{1}{6} + \frac{\sqrt{5}}{6} \right)}}{\log{\left(9 \right)}} = \log{\left(\left(- \frac{1}{6} + \frac{\sqrt{5}}{6}\right)^{\frac{1}{\log{\left(9 \right)}}} \right)}

w_{2} = \frac{\log{\left(- \frac{\sqrt{5}}{6} - \frac{1}{6} \right)}}{\log{\left(9 \right)}} = \frac{\log{\left(\frac{1}{6} + \frac{\sqrt{5}}{6} \right)} + i \pi}{\log{\left(9 \right)}}

hacemos cambio inverso

\cos{\left(x \right)} = w

Tenemos la ecuación

\cos{\left(x \right)} = w

es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en

x = \pi n + \operatorname{acos}{\left(w \right)}

x = \pi n + \operatorname{acos}{\left(w \right)} - \pi

x = \pi n + \operatorname{acos}{\left(w \right)}

x = \pi n + \operatorname{acos}{\left(w \right)} - \pi

, donde n es cualquier número entero
sustituimos w:

x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}

x_{1} = \pi n + \operatorname{acos}{\left(\log{\left(\left(- \frac{1}{6} + \frac{\sqrt{5}}{6}\right)^{\frac{1}{\log{\left(9 \right)}}} \right)} \right)}

x_{1} = \pi n + \operatorname{acos}{\left(\log{\left(\left(- \frac{1}{6} + \frac{\sqrt{5}}{6}\right)^{\frac{1}{\log{\left(9 \right)}}} \right)} \right)}

x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}

x_{2} = \pi n + \operatorname{acos}{\left(\frac{\log{\left(\frac{1}{6} + \frac{\sqrt{5}}{6} \right)} + i \pi}{\log{\left(9 \right)}} \right)}

x_{2} = \pi n + \operatorname{acos}{\left(\frac{\log{\left(\frac{1}{6} + \frac{\sqrt{5}}{6} \right)} + i \pi}{\log{\left(9 \right)}} \right)}

x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi

x_{3} = \pi n - \pi + \operatorname{acos}{\left(\log{\left(\left(- \frac{1}{6} + \frac{\sqrt{5}}{6}\right)^{\frac{1}{\log{\left(9 \right)}}} \right)} \right)}

x_{3} = \pi n - \pi + \operatorname{acos}{\left(\log{\left(\left(- \frac{1}{6} + \frac{\sqrt{5}}{6}\right)^{\frac{1}{\log{\left(9 \right)}}} \right)} \right)}

x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi

x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\log{\left(\frac{1}{6} + \frac{\sqrt{5}}{6} \right)} + i \pi}{\log{\left(9 \right)}} \right)}

x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\log{\left(\frac{1}{6} + \frac{\sqrt{5}}{6} \right)} + i \pi}{\log{\left(9 \right)}} \right)}

Gráfica

Trazar el gráfico f1(x)

Trazar el gráfico f2(x)

Suma y producto de raíces [src]

suma

    /    /-log(3) - pi*I\\              /    /-log(3) - pi*I\\       /    /-log(3) + pi*I\\              /    /-log(3) + pi*I\\       /    /-log(3) + pi*I\\     /    /-log(3) + pi*I\\       /    /-(pi*I + log(3)) \\     /    /-(pi*I + log(3)) \\
- re|acos|--------------|| + 2*pi - I*im|acos|--------------|| + - re|acos|--------------|| + 2*pi - I*im|acos|--------------|| + I*im|acos|--------------|| + re|acos|--------------|| + I*im|acos|-----------------|| + re|acos|-----------------||
    \    \   2*log(3)   //              \    \   2*log(3)   //       \    \   2*log(3)   //              \    \   2*log(3)   //       \    \   2*log(3)   //     \    \   2*log(3)   //       \    \     2*log(3)    //     \    \     2*log(3)    //

\left(\left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) + \left(\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right)\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right)

    /    /-log(3) - pi*I\\              /    /-(pi*I + log(3)) \\       /    /-log(3) - pi*I\\     /    /-(pi*I + log(3)) \\
- re|acos|--------------|| + 4*pi + I*im|acos|-----------------|| - I*im|acos|--------------|| + re|acos|-----------------||
    \    \   2*log(3)   //              \    \     2*log(3)    //       \    \   2*log(3)   //     \    \     2*log(3)    //

- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + 4 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}

producto

/    /    /-log(3) - pi*I\\              /    /-log(3) - pi*I\\\ /    /    /-log(3) + pi*I\\              /    /-log(3) + pi*I\\\ /    /    /-log(3) + pi*I\\     /    /-log(3) + pi*I\\\ /    /    /-(pi*I + log(3)) \\     /    /-(pi*I + log(3)) \\\
|- re|acos|--------------|| + 2*pi - I*im|acos|--------------|||*|- re|acos|--------------|| + 2*pi - I*im|acos|--------------|||*|I*im|acos|--------------|| + re|acos|--------------|||*|I*im|acos|-----------------|| + re|acos|-----------------|||
\    \    \   2*log(3)   //              \    \   2*log(3)   /// \    \    \   2*log(3)   //              \    \   2*log(3)   /// \    \    \   2*log(3)   //     \    \   2*log(3)   /// \    \    \     2*log(3)    //     \    \     2*log(3)    ///

\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right)

/    /    /-log(3) + pi*I\\     /    /-log(3) + pi*I\\\ /    /    /-log(3) - pi*I\\     /    /-log(3) - pi*I\\\ /            /    /-log(3) + pi*I\\     /    /-log(3) + pi*I\\\ /            /    /-log(3) - pi*I\\     /    /-log(3) - pi*I\\\
|I*im|acos|--------------|| + re|acos|--------------|||*|I*im|acos|--------------|| + re|acos|--------------|||*|-2*pi + I*im|acos|--------------|| + re|acos|--------------|||*|-2*pi + I*im|acos|--------------|| + re|acos|--------------|||
\    \    \   2*log(3)   //     \    \   2*log(3)   /// \    \    \   2*log(3)   //     \    \   2*log(3)   /// \            \    \   2*log(3)   //     \    \   2*log(3)   /// \            \    \   2*log(3)   //     \    \   2*log(3)   ///

\left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}\right)

(i*im(acos((-log(3) + pi*i)/(2*log(3)))) + re(acos((-log(3) + pi*i)/(2*log(3)))))*(i*im(acos((-log(3) - pi*i)/(2*log(3)))) + re(acos((-log(3) - pi*i)/(2*log(3)))))*(-2*pi + i*im(acos((-log(3) + pi*i)/(2*log(3)))) + re(acos((-log(3) + pi*i)/(2*log(3)))))*(-2*pi + i*im(acos((-log(3) - pi*i)/(2*log(3)))) + re(acos((-log(3) - pi*i)/(2*log(3)))))

Respuesta rápida [src]

         /    /-log(3) - pi*I\\              /    /-log(3) - pi*I\\
x1 = - re|acos|--------------|| + 2*pi - I*im|acos|--------------||
         \    \   2*log(3)   //              \    \   2*log(3)   //

x_{1} = - \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} - i \pi}{2 \log{\left(3 \right)}} \right)}\right)}

         /    /-log(3) + pi*I\\              /    /-log(3) + pi*I\\
x2 = - re|acos|--------------|| + 2*pi - I*im|acos|--------------||
         \    \   2*log(3)   //              \    \   2*log(3)   //

x_{2} = - \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}

         /    /-log(3) + pi*I\\     /    /-log(3) + pi*I\\
x3 = I*im|acos|--------------|| + re|acos|--------------||
         \    \   2*log(3)   //     \    \   2*log(3)   //

x_{3} = \operatorname{re}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{- \log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}

         /    /-(pi*I + log(3)) \\     /    /-(pi*I + log(3)) \\
x4 = I*im|acos|-----------------|| + re|acos|-----------------||
         \    \     2*log(3)    //     \    \     2*log(3)    //

x_{4} = \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{\log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{\log{\left(3 \right)} + i \pi}{2 \log{\left(3 \right)}} \right)}\right)}

x4 = re(acos(-(log(3) + i*pi)/(2*log(3)))) + i*im(acos(-(log(3) + i*pi)/(2*log(3))))

Respuesta numérica [src]

x1 = 4.42985922264678 - 1.18854986833522*i

x2 = 4.42985922264678 + 1.18854986833522*i

x3 = 1.85332608453281 - 1.18854986833522*i

x4 = 1.85332608453281 + 1.18854986833522*i

x4 = 1.85332608453281 + 1.18854986833522*i

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

9*81^cosx+3*9^cosx=0 la ecuación

Solución numérica:

Solución

981^cosx+39^cosx=0 la ecuación