Question

Si X⁴ + 1/X⁴ = 14 calcular: X + 1/X

Answer 1

Respuesta:

√6

Explicación paso a paso:

recordar :

$(x+\frac{1}{x} )^{2} = x^{2} +2.x\frac{1}{x} +\frac{1}{x^{2} } \\ \\ (x+\frac{1}{x} )^{2} =x^{2} +\frac{1}{x^{2} } +2$

elevando al cuadrado y agrupando :

$[(x+\frac{1}{x } )^{2}] ^{2} =[(x^{2} +\frac{1}{x^{2} } )+2]^{2} \\ \\ (x+\frac{1}{x } )^{4} =(x^{2} +\frac{1}{x^{2} } )^{2} +2.2(x^{2} +\frac{1}{x^{2} } )+4 \\ \\ (x+\frac{1}{x } )^{4}=x^{4} +\frac{1}{x^{4} } +2+4(x^{2} +\frac{1}{x^{2} } )+4\\ \\ (x+\frac{1}{x } )^{4}=14+2+4+4(x^{2} +\frac{1}{x^{2} } )\\ \\ (x+\frac{1}{x } )^{4}=20+4(x^{2} +\frac{1}{x^{2} })$

pero :

$x^{2}+ \frac{1}{x^{2} } =(x+\frac{1}{x})^{2} -2$

remplazando

$(x+\frac{1}{x})^{4} =20+4(x+\frac{1}{x} )^{2} -2)\\ \\ (x+\frac{1}{x})^{4} =20+4(x+\frac{1}{x} )^{2} -8\\ \\ (x+\frac{1}{x})^{4} =12+4(x+\frac{1}{x} )^{2}\\ \\ (x+\frac{1}{x})^{4}-4(x+\frac{1}{x} )^{2}-12=0\\ \\ ((x+\frac{1}{x} )^{2}-6)((x+\frac{1}{x} )^{2}+2) =0\\\\ (x+\frac{1}{x} )^{2}=6\\ \\ (x+\frac{1}{x} )=\sqrt{6}$

la otra opción de (x+1/x)² = -2 esta descartada

Saludos