Question

х3-1/х3=19(х-1/х) как решить урпвнение

Answer 1

$x^3- frac{1}{x^3}=19( x- frac{1}{x}) \ (x- frac{1}{x})(x^2+x*1/x+ frac{1}{x^2})=19( x- frac{1}{x}) \ x neq 0 \ x- frac{1}{x}=0|*x \ x^{2} -1=0=>x_1=1;x_2=-1 \ x^2+1+ frac{1}{x^2}=19 \ x^2+ frac{1}{x^2}=18|*x^2 \ x^4-18x^2+1=0 \ x^2=t \ t^2-18t+1=0 \ D=320=(8 sqrt{5} )^2 \ t_{1,2}= frac{18 pm 8 sqrt{5}}{2}=9 pm 4 sqrt{5}=2^2 pm 4 sqrt{5}+ sqrt{5}^2=(2 pm sqrt{5})^2 \ x^2= (2 pm sqrt{5})^2 =>x_{3,4}=2 pm sqrt{5}$