HMMT 二月 2004 · 代数 · 第 6 题
HMMT February 2004 — Algebra — Problem 6
题目详情
- Find all real solutions to x + (2 − x ) = 34 .
解析
- Find all real solutions to x + (2 − x ) = 34 . √ Solution: 1 ± 2 4 4 Let y = 2 − x, so x + y = 2 and x + y = 34 . We know 4 4 3 2 2 3 4 4 4 2 2 ( x + y ) = x + 4 x y + 6 x y + 4 xy + y = x + y + 2 xy (2 x + 2 y + 3 xy ) . 2 2 2 4 Moreover, x + y = ( x + y ) − 2 xy , so the preceding equation becomes 2 = 34+2 xy (2 · 2 2 2 − xy ) , or ( xy ) − 8 xy − 9 = 0 . Hence xy = 9 or − 1 . Solving xy = 9 , x + y = 2 produces √ √ complex solutions, and solving xy = − 1 , x + y = 2 produces ( x, y ) = (1 + 2 , 1 − 2) √ √ √ or (1 − 2 , 1 + 2) . Thus, x = 1 ± 2 .