Question

若x+x^-1=3，分别求代数x^2+x^-2和x^2-x^-2的值

2：log（4）9-log（5）4xlog（8）5
log（4）9-log（2）12+log（5）4xlog（8）5

Answer 1

x+x^-1=3
(x+x^-1)^2=x^2+x^-2+2=9
∴x^2+x^-2=7
x^2+x^-2-2*x*x^-1=(x-x^-1)^2=5
∴x-x^-1=±√5
∴x^2-x^-2=(x+x^-1)(x-x^-1)=±3√5
(2)原式=log(2)3-(log(2)3+log(2)4)+2log(5)2×1/3log(2)5
=-2+2/3
=-4/3

Answer 2

解：已知x+1/x=3,则(x+1/x)^2=9,即：
x^2+x^-2=9-2=7
x-1/x=√(x-1/x)^2=√(x^2-2*x*1/x+x^-2)=√5
x^2-x^-2=(x+1/x)(x-1/x)=3*√5=3√5

log（4）9-log（5）4xlog（8）5
=log(2^2)3^2-log(5)2^2*log(2^3)5
=log(2)3-2/3*log(5)2*log(2)5
=log(2)3-2/3