x^2+1/x^2+x+1/x=0
x^2+1/x^2+x+1/x=0
(x+1/x)^2+(x+1/x)-2=0
令T=x+1/x
=>T^2+T-2=0
(T-1)(T+2)=0.
T=1或者T=-2.
=>x+1/x=1或者x+1/x=-2.
(1)x+1/x=1时.
x^2+1-x=0
=>x^2-x+1/4=-3/4
(x-1/2)^2=-3/4.
=>显然不成立,也就是说,不存在这样的x.
(2)x+1/x=-2时.
x^2+1+2x=0
(x+1)^2=0
=>x=-1.