作者ejialan (eji)
看板Math
标题Re: [中学] 高中数学一题
时间Wed Sep 30 12:31:24 2020
※ 引述《adamchi (adamchi)》之铭言：
: n=2,4,6,8 时,
: ((x^2)/(n^2-1^2)) +((y^2)/(n^2-3^2))+((z^2)/(n^2-5^2))+((w^2)/(n^2-7^2)) = 1
: ,求x^2+y^2+z^2+w^2 = ?
: 本题以计算机暴力解法得答案为36,
: 想请教有何更快的解法,谢谢~
令u=n^2，两边同乘(u-1)(u-9)(u-25)(u-49)可得
x^2(u-9)(u-25)(u-49)+y^2(u-1)(u-25)(u-49)
+z^2(u-1)(u-9)(u-49)+w^2(u-1)(u-9)(u-25) = (u-1)(u-9)(u-25)(u-49)
移项整理为u的4次多项式，4根为4,16,36,64
4次方项系数为-1，3次方项系数为x^2+y^2+z^2+w^2+84 (84=1+9+25+49)
由根与系数关系得x^2+y^2+z^2+w^2+84 = 4+16+36+64
=> x^2+y^2+z^2+w^2 = 36
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