x1+x2=3k-1
x1x2=k²+4
则1/x1+1/x2
=(x1+x2)/x1x2
=(3k-1)/(k²+4)=5/8
5k²-24k+28=0
(5k-14)(k-2)=0
k=14/5,k=2
△>=0
9k²-6k+1-4k²-16>=0
5k²-6k-15>=0
k=2舍去
k=14/5