Question

x^3+6+1/x^3=2(x+1/x)^2 求x+1/x

Answer 1

令a=x+1/x

则a²=x²+2+1/x²

x²+1/x²=a²-2

x³+1/x³

=(x+1/x)(x²-1+1/x²)

=a(a²-2-1)

=a³-3a

所以即a³-3a+6=2a²

a³-2a²-3a+6=0

a²(a-2)-3(a-2)=0

(a-2)(a²-3)=0

a=2,a=±√3

x+1/x=±√3时

即x²±√3x+1=0

此时判别式小于0

没有实数解

所以x+1/x=2