HMMT 二月 2004 · 代数 · 第 6 题

英文解析

6. 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 .