HMMT 十一月 2008 · GEN1 赛 · 第 9 题

英文解析

9. [ 7 ] Find the product of all real x for which
   3 x +1 2 x x +3
   2 − 17 · 2 + 2 = 0 .
   x x 2 x x 2 x
   Answer: -3 We can re-write the equation as 2 (2 · (2 ) − 17 · (2 )+8) = 0, or 2 · (2 ) − 17 · (2 )+8 = 0.
   x 2
   Make the substitution y = 2 . Then we have 2 y − 17 y + 8 = 0, which has solutions (by the quadratic
   √
   17 ± 289 − 64
   17 ± 15 1 x 1
   formula) y = = = 8 , , so 2 = 8 , and x = 3 , − 1. The product of these numbers is
   4 4 2 2 − 3.
   3 2