题目

x²-3x+1=0,根为x1x2
①x1³+x2³ ②1/x1²-1/x2² ③(x1+1/x1)(x2+1/x2)

提问时间：2021-04-01

答案

x^2-3x+1=0 ==> x1=3/2+√5/2 ,x2=3/2-√5/2 ==> x1+x2=3 x1x2=1
x^2=3x-1
1.x1^3+x2^3
= （x1^2）x1 + (x2^2)x2
=(3x1-1)x1 + (3x2-1)x2
= 3x1^2-x1+3x2^2-x2
= 3(3x1-1)-x1 + 3(3x2-1) -x2
= 9x1-3-x1+9x2-3-x2
= 8(x1+x2)-6=8*3-6=18
2.1/x1²-1/x2²
= (x1²+x2²)/(x1²x2²)
=(3x1-1+3x2-1)/((3x1-1)(3x2-1))
= 7/(9x1x2-3(x1+x2)+1)
=7/(9*1-9+1)=7
3.(x1+1/x1)(x2+1/x2)
= ((x1²+1)/x1)*((x2²+1)/x2)
= ((3x1-1+1)/x1)*((3x2-1+1)/x2)
=(3x1/x1)*(3x2/x2)
=3*3=9