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sqrtx^3-4*x/x^2=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

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Solución numérica:

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Solución

Ha introducido [src] 
     3          
  ___    4*x    
\/ x   - --- = 0
           2    
          x
(x)3−4xx2=0\left(\sqrt{x}\right)^{3} - \frac{4 x}{x^{2}} = 0
Simplificar
Solución detallada
Tenemos la ecuación
(x)3−4xx2=0\left(\sqrt{x}\right)^{3} - \frac{4 x}{x^{2}} = 0
cambiamos
x52=4x^{\frac{5}{2}} = 4
Ya que la potencia en la ecuación es igual a = 5/2 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Elevemos las dos partes de la ecuación a la potencia 2/5:
Obtenemos:
(x52)25=425\left(x^{\frac{5}{2}}\right)^{\frac{2}{5}} = 4^{\frac{2}{5}}
o
x=245x = 2^{\frac{4}{5}}
Abrimos los paréntesis en el miembro derecho de la ecuación
x = 2^4/5

Obtenemos la respuesta: x = 2^(4/5)

Las demás 4 raíces son complejas.
hacemos el cambio:
z=xz = x
entonces la ecuación será así:
z52=4z^{\frac{5}{2}} = 4
Cualquier número complejo se puede presentar que:
z=reipz = r e^{i p}
sustituimos en la ecuación
(reip)52=4\left(r e^{i p}\right)^{\frac{5}{2}} = 4
donde
r=245r = 2^{\frac{4}{5}}
- módulo del número complejo
Sustituyamos r:
e5ip2=1e^{\frac{5 i p}{2}} = 1
Usando la fórmula de Euler hallemos las raíces para p
isin⁡(5p2)+cos⁡(5p2)=1i \sin{\left(\frac{5 p}{2} \right)} + \cos{\left(\frac{5 p}{2} \right)} = 1
es decir
cos⁡(5p2)=1\cos{\left(\frac{5 p}{2} \right)} = 1
y
sin⁡(5p2)=0\sin{\left(\frac{5 p}{2} \right)} = 0
entonces
p=4πN5p = \frac{4 \pi N}{5}
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
z1=245z_{1} = 2^{\frac{4}{5}}
z2=(−2254+22554−225i58+58)2z_{2} = \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - 2^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)^{2}
z3=(−2254+22554+225i58+58)2z_{3} = \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4} + 2^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)^{2}
z4=(−22554−2254−225i58−58)2z_{4} = \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4} - 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)^{2}
z5=(−22554−2254+225i58−58)2z_{5} = \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4} + 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)^{2}
hacemos cambio inverso
z=xz = x
x=zx = z

Entonces la respuesta definitiva es:
x1=245x_{1} = 2^{\frac{4}{5}}
x2=(−2254+22554−225i58+58)2x_{2} = \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - 2^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)^{2}
x3=(−2254+22554+225i58+58)2x_{3} = \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4} + 2^{\frac{2}{5}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)^{2}
x4=(−22554−2254−225i58−58)2x_{4} = \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4} - 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)^{2}
x5=(−22554−2254+225i58−58)2x_{5} = \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4} + 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)^{2}
Gráfica
-10.0-7.5-5.0-2.50.02.55.07.510.012.515.017.5-1000010000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
      4/5
x1 = 2
x1=245x_{1} = 2^{\frac{4}{5}} 
                          2                                    ___________                      
     /   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\
     |  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 |
x2 = |- ---- + ----------|  - 2   *|- + -----| - 2*I*2   *  /   - + ----- *|- ---- + ----------|
     \   4         4     /         \8     8  /            \/    8     8    \   4         4     /
x2=−245(58+58)+(−2254+22554)2−2⋅225i(−2254+22554)58+58x_{2} = - 2^{\frac{4}{5}} \left(\frac{\sqrt{5}}{8} + \frac{5}{8}\right) + \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right)^{2} - 2 \cdot 2^{\frac{2}{5}} i \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right) \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} 
                          2                                    ___________                      
     /   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\
     |  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 |
x3 = |- ---- + ----------|  - 2   *|- + -----| + 2*I*2   *  /   - + ----- *|- ---- + ----------|
     \   4         4     /         \8     8  /            \/    8     8    \   4         4     /
x3=−245(58+58)+(−2254+22554)2+2⋅225i(−2254+22554)58+58x_{3} = - 2^{\frac{4}{5}} \left(\frac{\sqrt{5}}{8} + \frac{5}{8}\right) + \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right)^{2} + 2 \cdot 2^{\frac{2}{5}} i \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right) \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} 
                          2                                    ___________                      
     /   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\
     |  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 |
x4 = |- ---- - ----------|  - 2   *|- - -----| - 2*I*2   *  /   - - ----- *|- ---- - ----------|
     \   4         4     /         \8     8  /            \/    8     8    \   4         4     /
x4=−245(58−58)+(−22554−2254)2−2⋅225i58−58(−22554−2254)x_{4} = - 2^{\frac{4}{5}} \left(\frac{5}{8} - \frac{\sqrt{5}}{8}\right) + \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)^{2} - 2 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right) 
                          2                                    ___________                      
     /   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\
     |  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 |
x5 = |- ---- - ----------|  - 2   *|- - -----| + 2*I*2   *  /   - - ----- *|- ---- - ----------|
     \   4         4     /         \8     8  /            \/    8     8    \   4         4     /
x5=−245(58−58)+(−22554−2254)2+2⋅225i58−58(−22554−2254)x_{5} = - 2^{\frac{4}{5}} \left(\frac{5}{8} - \frac{\sqrt{5}}{8}\right) + \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)^{2} + 2 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right) 
x5 = -2^(4/5)*(5/8 - sqrt(5)/8) + (-2^(2/5)*sqrt(5)/4 - 2^(2/5)/4)^2 + 2*2^(2/5)*i*sqrt(5/8 - sqrt(5)/8)*(-2^(2/5)*sqrt(5)/4 - 2^(2/5)/4)
Suma y producto de raíces [src] 
suma
                            2                                    ___________                                              2                                    ___________                                              2                                    ___________                                              2                                    ___________                      
       /   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\   /   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\   /   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\   /   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\
 4/5   |  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 |   |  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 |   |  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 |   |  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 |
2    + |- ---- + ----------|  - 2   *|- + -----| - 2*I*2   *  /   - + ----- *|- ---- + ----------| + |- ---- + ----------|  - 2   *|- + -----| + 2*I*2   *  /   - + ----- *|- ---- + ----------| + |- ---- - ----------|  - 2   *|- - -----| - 2*I*2   *  /   - - ----- *|- ---- - ----------| + |- ---- - ----------|  - 2   *|- - -----| + 2*I*2   *  /   - - ----- *|- ---- - ----------|
       \   4         4     /         \8     8  /            \/    8     8    \   4         4     /   \   4         4     /         \8     8  /            \/    8     8    \   4         4     /   \   4         4     /         \8     8  /            \/    8     8    \   4         4     /   \   4         4     /         \8     8  /            \/    8     8    \   4         4     /
(−245(58−58)+(−22554−2254)2+2⋅225i58−58(−22554−2254))+(((245+(−245(58+58)+(−2254+22554)2−2⋅225i(−2254+22554)58+58))+(−245(58+58)+(−2254+22554)2+2⋅225i(−2254+22554)58+58))+(−245(58−58)+(−22554−2254)2−2⋅225i58−58(−22554−2254)))\left(- 2^{\frac{4}{5}} \left(\frac{5}{8} - \frac{\sqrt{5}}{8}\right) + \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)^{2} + 2 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)\right) + \left(\left(\left(2^{\frac{4}{5}} + \left(- 2^{\frac{4}{5}} \left(\frac{\sqrt{5}}{8} + \frac{5}{8}\right) + \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right)^{2} - 2 \cdot 2^{\frac{2}{5}} i \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right) \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) + \left(- 2^{\frac{4}{5}} \left(\frac{\sqrt{5}}{8} + \frac{5}{8}\right) + \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right)^{2} + 2 \cdot 2^{\frac{2}{5}} i \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right) \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) + \left(- 2^{\frac{4}{5}} \left(\frac{5}{8} - \frac{\sqrt{5}}{8}\right) + \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)^{2} - 2 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)\right)\right) 
=
                              2                          2                                          
         /   2/5    2/5   ___\      /   2/5    2/5   ___\           /      ___\          /      ___\
 4/5     |  2      2   *\/ 5 |      |  2      2   *\/ 5 |       4/5 |5   \/ 5 |      4/5 |5   \/ 5 |
2    + 2*|- ---- - ----------|  + 2*|- ---- + ----------|  - 2*2   *|- - -----| - 2*2   *|- + -----|
         \   4         4     /      \   4         4     /           \8     8  /          \8     8  /
−2⋅245(58+58)−2⋅245(58−58)+2(−2254+22554)2+245+2(−22554−2254)2- 2 \cdot 2^{\frac{4}{5}} \left(\frac{\sqrt{5}}{8} + \frac{5}{8}\right) - 2 \cdot 2^{\frac{4}{5}} \left(\frac{5}{8} - \frac{\sqrt{5}}{8}\right) + 2 \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right)^{2} + 2^{\frac{4}{5}} + 2 \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)^{2} 
producto
     /                     2                                    ___________                      \ /                     2                                    ___________                      \ /                     2                                    ___________                      \ /                     2                                    ___________                      \
     |/   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\| |/   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\| |/   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\| |/   2/5    2/5   ___\         /      ___\                /       ___  /   2/5    2/5   ___\|
 4/5 ||  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 || ||  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 || ||  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 || ||  2      2   *\/ 5 |     4/5 |5   \/ 5 |        2/5    /  5   \/ 5   |  2      2   *\/ 5 ||
2   *||- ---- + ----------|  - 2   *|- + -----| - 2*I*2   *  /   - + ----- *|- ---- + ----------||*||- ---- + ----------|  - 2   *|- + -----| + 2*I*2   *  /   - + ----- *|- ---- + ----------||*||- ---- - ----------|  - 2   *|- - -----| - 2*I*2   *  /   - - ----- *|- ---- - ----------||*||- ---- - ----------|  - 2   *|- - -----| + 2*I*2   *  /   - - ----- *|- ---- - ----------||
     \\   4         4     /         \8     8  /            \/    8     8    \   4         4     // \\   4         4     /         \8     8  /            \/    8     8    \   4         4     // \\   4         4     /         \8     8  /            \/    8     8    \   4         4     // \\   4         4     /         \8     8  /            \/    8     8    \   4         4     //
245(−245(58+58)+(−2254+22554)2−2⋅225i(−2254+22554)58+58)(−245(58+58)+(−2254+22554)2+2⋅225i(−2254+22554)58+58)(−245(58−58)+(−22554−2254)2−2⋅225i58−58(−22554−2254))(−245(58−58)+(−22554−2254)2+2⋅225i58−58(−22554−2254))2^{\frac{4}{5}} \left(- 2^{\frac{4}{5}} \left(\frac{\sqrt{5}}{8} + \frac{5}{8}\right) + \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right)^{2} - 2 \cdot 2^{\frac{2}{5}} i \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right) \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(- 2^{\frac{4}{5}} \left(\frac{\sqrt{5}}{8} + \frac{5}{8}\right) + \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right)^{2} + 2 \cdot 2^{\frac{2}{5}} i \left(- \frac{2^{\frac{2}{5}}}{4} + \frac{2^{\frac{2}{5}} \sqrt{5}}{4}\right) \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right) \left(- 2^{\frac{4}{5}} \left(\frac{5}{8} - \frac{\sqrt{5}}{8}\right) + \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)^{2} - 2 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)\right) \left(- 2^{\frac{4}{5}} \left(\frac{5}{8} - \frac{\sqrt{5}}{8}\right) + \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)^{2} + 2 \cdot 2^{\frac{2}{5}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \left(- \frac{2^{\frac{2}{5}} \sqrt{5}}{4} - \frac{2^{\frac{2}{5}}}{4}\right)\right) 
=
16
1616 
16
Respuesta numérica [src] 
x1 = 1.74110112659225
x2 = -1.4085804003385 - 1.02339356496073*i
x3 = -1.4085804003385 + 1.02339356496073*i
x4 = 0.538029837042371 + 1.65588557197439*i
x5 = 0.538029837042371 - 1.65588557197439*i
x5 = 0.538029837042371 - 1.65588557197439*i
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