Question

已知x/（x^2-x+1）=1/3,求x^2/（x^4+x^2+1）的值

Answer 1

x/（x^2-x+1）=1/3
∴(x^2-x+1)/x=3
x-1+1/x=3
x+1/x=4
两边平方得
x^2+2+1/x^2=16
x^2+1+1/x^2=15
(X^4+X^2+1)/X^2=15
∴x^2/（x^4+x^2+1）=1/15

Answer 2

解：因为 x/(x^2--x+1)=1/3,
所以 (x^2--x+1)/x=3,
即： x--1+1/x=3,
x+1/x=4,
两边平方得：
x^2+2+1/x^2=16,
即： x^2+1+1/x^2=15
(x^4+x^2+1)/x^2=15,
所以 x^2/(x^4+x^2+1)/x^2=1/15。

Answer 3

解法一：
∵x/(x^2-x+1)=1/3,
∴3x=x^2-x+1
x^2=4x-1
x^4=16x^2-8x+1
x^2/（x^4+x^2+1）
=(4x-1)/(16x^2-8x+1+x^2+1)
=(4x-1)/(17x^2-8x+2)
=(4x-1)/(17(4x-1)-8x+2)
=(4x-1)/(68x-17)-8x+2)
=(4x-1)/(60x-15)
=1/15(4x-1)/(4x-1)
=1/15
解法二：
∵x/(x^2-x+1)=1/3,
∴3x=x^2-x+1
x+1/x=4
x^2+2+1/x^2=16
x^2+1+1/x^2=15
(x^4+x^2+1)/x^2=15,
∴ x^2/(x^4+x^2+1)=1/15。

Answer 4

x/（x^2-x+1）=1/3,
1/(x+1/x-1)=1/3
x+1/x=4
(x+1/x)²=16
x²+1/x²=14

x^2/（x^4+x^2+1）
=1/(x²+1/x²+1）
=1/15
记住(a+1/a)²=a²+1/a²+2