题目

已知a=1/2+根号3,求代数式a平方-1/a+1-a平方-2a+1/a平方-a的值

提问时间：2020-07-27

答案

a=1/(2+√3)
=(2-√3)/[(2-√3)(2+√3)]
=(2-√3)/(2²-√3²)
=(2-√3)/(4-3)
=2-√3
原式=[(a²-1)/(a+1)]-[(a²-2a+1)/(a²-a)]
=[(a+1)(a-1)/(a+1)]-{ (a-1)²/[a(a-1)] }
=(a-1)-[(a-1)/a]
=[a(a-1)-(a-1)]/a
=(a²-a-a+1)/a
=(a²-2a+1)/a
=(a-1)²/a
当a=2-√3时
原式=(2-√3-1)²/(2-√3)
=(1-√3)²(2+√3)/[(2+√3)(2-√3)]
=(1-2√3+3)(2+√3)/(2²-√3²)
=(4-2√3)(2+√3)/(4-3)
=4×2+4×√3-2√3×2-2√3×√3
=8+4√3-4√3-6
=2