HMMT 二月 2001 · 代数 · 第 10 题

英文解析

10. Find the real solutions of (2 x + 1)(3 x + 1)(5 x + 1)(30 x + 1) = 10.
    Solution: (2 x + 1)(3 x + 1)(5 x + 1)(30 x + 1) = [(2 x + 1)(30 x + 1)][(3 x + 1)(5 x + 1)] =
    2 2 2
    (60 x + 32 x + 1)(15 x + 8 x + 1) = (4 y + 1)( y + 1) = 10, where y = 15 x + 8 x . The quadratic
    √
    9 − 4 ± 31
    equation in y yields y = 1 and y = − . For y = 1, we have 15 x + 8 x − 1 = 0, so x = .2
    4 15
    9 9
    For y = − , we have 15 x + 8 x + , which yields only complex solutions for x . Thus the real 2
    4 4
    √
    − 4 ± 31
    solutions are .
    15