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Expresiones semejantes
- ((2/3)^5x^2+29-(3/2)^x62-5)*√(4x+7)=0
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- ((2/3)^5x^2-29-(3/2)^x62-5)*√(4x-7)=0
((2/3)^5x^2-29-(3/2)^x62-5)*√(4x+7)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
/ 5 2 x62 \ _________ \2/3 *x - 29 - 3/2 - 5/*\/ 4*x + 7 = 0
$$\sqrt{4 x + 7} \left(\left(- \left(\frac{3}{2}\right)^{x_{62}} + \left(\left(\frac{2}{3}\right)^{5} x^{2} - 29\right)\right) - 5\right) = 0$$
Solución detallada
Tenemos la ecuación:
$$\sqrt{4 x + 7} \left(\left(- \left(\frac{3}{2}\right)^{x_{62}} + \left(\left(\frac{2}{3}\right)^{5} x^{2} - 29\right)\right) - 5\right) = 0$$
Ya que la parte derecha de la ecuación es igual a cero, la solución de la ecuación será, si por lo menos uno de los factores en la parte izquierda de la ecuación es igual a cero.
Obtenemos ecuaciones
$$4 x + 7 = 0$$
$$- \left(\frac{3}{2}\right)^{x_{62}} + \frac{32 x^{2}}{243} - 34 = 0$$
resolvemos las ecuaciones obtenidas:
1.
$$4 x + 7 = 0$$
Transportamos los términos libres (sin x)
del miembro izquierdo al derecho, obtenemos:
$$4 x = -7$$
Dividamos ambos miembros de la ecuación en 4
Obtenemos la respuesta: x1 = -7/4
2.
$$- \left(\frac{3}{2}\right)^{x_{62}} + \frac{32 x^{2}}{243} - 34 = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = \frac{32}{243}$$
$$b = 0$$
$$c = - \left(\frac{3}{2}\right)^{x_{62}} - 34$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{2} = \frac{243 \sqrt{\frac{128 \left(\frac{3}{2}\right)^{x_{62}}}{243} + \frac{4352}{243}}}{64}$$
$$x_{3} = - \frac{243 \sqrt{\frac{128 \left(\frac{3}{2}\right)^{x_{62}}}{243} + \frac{4352}{243}}}{64}$$
Entonces la respuesta definitiva es:
$$x_{1} = - \frac{7}{4}$$
$$x_{2} = \frac{243 \sqrt{\frac{128 \left(\frac{3}{2}\right)^{x_{62}}}{243} + \frac{4352}{243}}}{64}$$
$$x_{3} = - \frac{243 \sqrt{\frac{128 \left(\frac{3}{2}\right)^{x_{62}}}{243} + \frac{4352}{243}}}{64}$$
$$\sqrt{4 x + 7} \left(\left(- \left(\frac{3}{2}\right)^{x_{62}} + \left(\left(\frac{2}{3}\right)^{5} x^{2} - 29\right)\right) - 5\right) = 0$$
Ya que la parte derecha de la ecuación es igual a cero, la solución de la ecuación será, si por lo menos uno de los factores en la parte izquierda de la ecuación es igual a cero.
Obtenemos ecuaciones
$$4 x + 7 = 0$$
$$- \left(\frac{3}{2}\right)^{x_{62}} + \frac{32 x^{2}}{243} - 34 = 0$$
resolvemos las ecuaciones obtenidas:
1.
$$4 x + 7 = 0$$
Transportamos los términos libres (sin x)
del miembro izquierdo al derecho, obtenemos:
$$4 x = -7$$
Dividamos ambos miembros de la ecuación en 4
x = -7 / (4)
Obtenemos la respuesta: x1 = -7/4
2.
$$- \left(\frac{3}{2}\right)^{x_{62}} + \frac{32 x^{2}}{243} - 34 = 0$$
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:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = \frac{32}{243}$$
$$b = 0$$
$$c = - \left(\frac{3}{2}\right)^{x_{62}} - 34$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (32/243) * (-34 - (3/2)^x62) = 4352/243 + 128*(3/2)^x62/243
La ecuación tiene dos raíces.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
o
$$x_{2} = \frac{243 \sqrt{\frac{128 \left(\frac{3}{2}\right)^{x_{62}}}{243} + \frac{4352}{243}}}{64}$$
$$x_{3} = - \frac{243 \sqrt{\frac{128 \left(\frac{3}{2}\right)^{x_{62}}}{243} + \frac{4352}{243}}}{64}$$
Entonces la respuesta definitiva es:
$$x_{1} = - \frac{7}{4}$$
$$x_{2} = \frac{243 \sqrt{\frac{128 \left(\frac{3}{2}\right)^{x_{62}}}{243} + \frac{4352}{243}}}{64}$$
$$x_{3} = - \frac{243 \sqrt{\frac{128 \left(\frac{3}{2}\right)^{x_{62}}}{243} + \frac{4352}{243}}}{64}$$
Gráfica
Suma y producto de raíces
[src]
suma
__________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\ __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\ __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\ __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\
/ 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|| / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|| / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|| / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /||
/ / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------|| / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------|| / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------|| / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------||
7 / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /| / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /| / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /| / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /|
- - + - 4 / |---- + --------------------| + ----------------------- *cos|--------------------------------------------------------| - I*4 / |---- + --------------------| + ----------------------- *sin|--------------------------------------------------------| + 4 / |---- + --------------------| + ----------------------- *cos|--------------------------------------------------------| + I*4 / |---- + --------------------| + ----------------------- *sin|--------------------------------------------------------|
4 \/ \ 16 32 / 1024 \ 2 / \/ \ 16 32 / 1024 \ 2 / \/ \ 16 32 / 1024 \ 2 / \/ \ 16 32 / 1024 \ 2 /
$$\left(i \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)} + \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)}\right) + \left(\left(- i \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)} - \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)}\right) - \frac{7}{4}\right)$$
=
-7/4
$$- \frac{7}{4}$$
producto
/ __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\ __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\\
| / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|| / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|||
| / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------|| / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------||| / __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\ __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\\
| / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /| / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /|| | / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|| / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|||
-7*|- 4 / |---- + --------------------| + ----------------------- *cos|--------------------------------------------------------| - I*4 / |---- + --------------------| + ----------------------- *sin|--------------------------------------------------------|| | / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------|| / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------|||
\ \/ \ 16 32 / 1024 \ 2 / \/ \ 16 32 / 1024 \ 2 // | / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /| / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /||
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------*|4 / |---- + --------------------| + ----------------------- *cos|--------------------------------------------------------| + I*4 / |---- + --------------------| + ----------------------- *sin|--------------------------------------------------------||
4 \\/ \ 16 32 / 1024 \ 2 / \/ \ 16 32 / 1024 \ 2 //
$$- \frac{7 \left(- i \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)} - \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)}\right)}{4} \left(i \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)} + \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)}\right)$$
=
/ log(3)*re(x62) - log(2)*re(x62) log(3)*re(x62) - log(2)*re(x62)\
__________________________________________ |-243*e *sin(im(x62)*log(2) - im(x62)*log(3)) 4131 243*cos(im(x62)*log(2) - im(x62)*log(3))*e |
/ 2 I*atan2|--------------------------------------------------------------------------, ---- + -------------------------------------------------------------------------|
/ / / -x62 x62\\ 2/ -x62 x62\ \ 32 16 32 /
1701*\/ \34 + re\2 *3 // + im \2 *3 / *e
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
128
$$\frac{1701 \sqrt{\left(\operatorname{re}{\left(2^{- x_{62}} \cdot 3^{x_{62}}\right)} + 34\right)^{2} + \left(\operatorname{im}{\left(2^{- x_{62}} \cdot 3^{x_{62}}\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- \frac{243 e^{- \log{\left(2 \right)} \operatorname{re}{\left(x_{62}\right)} + \log{\left(3 \right)} \operatorname{re}{\left(x_{62}\right)}} \sin{\left(- \log{\left(3 \right)} \operatorname{im}{\left(x_{62}\right)} + \log{\left(2 \right)} \operatorname{im}{\left(x_{62}\right)} \right)}}{32},\frac{243 e^{- \log{\left(2 \right)} \operatorname{re}{\left(x_{62}\right)} + \log{\left(3 \right)} \operatorname{re}{\left(x_{62}\right)}} \cos{\left(- \log{\left(3 \right)} \operatorname{im}{\left(x_{62}\right)} + \log{\left(2 \right)} \operatorname{im}{\left(x_{62}\right)} \right)}}{32} + \frac{4131}{16} \right)}}}{128}$$
1701*sqrt((34 + re(2^(-x62)*3^x62))^2 + im(2^(-x62)*3^x62)^2)*exp(i*atan2(-243*exp(log(3)*re(x62) - log(2)*re(x62))*sin(im(x62)*log(2) - im(x62)*log(3))/32, 4131/16 + 243*cos(im(x62)*log(2) - im(x62)*log(3))*exp(log(3)*re(x62) - log(2)*re(x62))/32))/128
Respuesta rápida
[src]
x1 = -7/4
$$x_{1} = - \frac{7}{4}$$
__________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\ __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\
/ 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|| / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /||
/ / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------|| / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------||
/ |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /| / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /|
x2 = - 4 / |---- + --------------------| + ----------------------- *cos|--------------------------------------------------------| - I*4 / |---- + --------------------| + ----------------------- *sin|--------------------------------------------------------|
\/ \ 16 32 / 1024 \ 2 / \/ \ 16 32 / 1024 \ 2 /
$$x_{2} = - i \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)} - \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)}$$
__________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\ __________________________________________________________ / / / -2*x62 x62\ / -2*x62 x62\\\
/ 2 | |243*im\2 *6 / 4131 243*re\2 *6 /|| / 2 | |243*im\2 *6 / 4131 243*re\2 *6 /||
/ / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------|| / / / -2*x62 x62\\ 2/ -2*x62 x62\ |atan2|--------------------, ---- + --------------------||
/ |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /| / |4131 243*re\2 *6 /| 59049*im \2 *6 / | \ 32 16 32 /|
x3 = 4 / |---- + --------------------| + ----------------------- *cos|--------------------------------------------------------| + I*4 / |---- + --------------------| + ----------------------- *sin|--------------------------------------------------------|
\/ \ 16 32 / 1024 \ 2 / \/ \ 16 32 / 1024 \ 2 /
$$x_{3} = i \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)} + \sqrt[4]{\left(\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16}\right)^{2} + \frac{59049 \left(\operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}\right)^{2}}{1024}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{243 \operatorname{im}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32},\frac{243 \operatorname{re}{\left(2^{- 2 x_{62}} \cdot 6^{x_{62}}\right)}}{32} + \frac{4131}{16} \right)}}{2} \right)}$$
x3 = i*((243*re(2^(-2*x62)*6^x62)/32 + 4131/16)^2 + 59049*im(2^(-2*x62)*6^x62)^2/1024)^(1/4)*sin(atan2(243*im(2^(-2*x62)*6^x62/32, 243*re(2^(-2*x62)*6^x62)/32 + 4131/16)/2) + ((243*re(2^(-2*x62)*6^x62)/32 + 4131/16)^2 + 59049*im(2^(-2*x62)*6^x62)^2/1024)^(1/4)*cos(atan2(243*im(2^(-2*x62)*6^x62)/32, 243*re(2^(-2*x62)*6^x62)/32 + 4131/16)/2))