Question

已知X=(4-根号7)/3, 那么求x2/(x4+x2+1) 的值

Answer 1

x=（4-√7）/3
x²=(16+7-8√7)/9=(32-8√7)/9-9/9
所以
x²=(8/3)x-1
即x²+1=(8/3)x
所以 x+1/x=8/3
(x+1/x)²=64/9
x²+1/x²+2=64/9
所以
x2/(x4+x2+1)
=1/(x²+1+1/x²)
=1/(64/9-1)
=1/(55/9)
=9/55

Answer 2

(x^4+x^2+1)/x^2
=x^2+1+1/x^2
=(1+1/x)^1-1
=[1+3/(4-根号7)]^2-1
=[1+(4+根号7)/3]^2-1
=(7+根号7)^2/9-1
=(47+14根号7)/9
x2/(x4+x2+1)
=9/(47+14根号7)
=（47-14根号7）/93

Answer 3

X=(4-√7)/3是方程3x²-8x+3=0的1个根
∴3x²+3=8x
∴x+1/x=8/3
∴(x+1/x)²=64/9
∴x²+1/x²+2=64/9
∴x²+1/x²=46/9
∴x²/(x⁴+x²+1)
=1/(x²+1/x²+1)
=1/(46/9+1)
=9/55