若a是方程x²+3x-1=0的根,求a²+1/a²

问题描述:

若a是方程x²+3x-1=0的根,求a²+1/a²

=(a-1/a)²+2=(-3)²+2=11
a²+3a-1=0
a+3-1/a=0
a-1/a=-3

按下面的方法算还不如直接求出X的值代入来的快

a²+1/a²=(a^4+1)/a²
=(a²(a²+3a-1)-3a^3+a²+1)/a²
=(a²(a²+3a-1)-3a(a²+3a-1)+9a²-3a+a²+1)/a²
=(a²(a²+3a-1)-3a(a²+3a-1)+11a²-(a²+3a-1))/a²
=(0-0+11a²-0)/a²
=11