ab不等于0,说明a,b都不等于0.
3a^2 + ab + 2b^2 = 0 等价于
3(a/b)^2 + (a/b) + 2 = 0.
记t = a/b
则
3t^2 + t + 2 = 0.
1^2 - 4*3*2 = - 23 < 0.
方程无解.
所以,不存在非零实数a,b能使得 3a^2 + ab + 2b^2 = 0成立.
若题目是
3a^2 + ab - 2b^2 = 0,求a/b - b/a - (a^2 + b^2)/(ab).
解法如下
ab不等于0,说明a,b都不等于0.
a/b - b/a - (a^2 + b^2)/(ab)
= a/b - b/a - a/b - b/a
= -2b/a
3a^2 + ab - 2b^2 = 0 等价于
2(b/a)^2 - (b/a) - 3 = 0.
记t = b/a
则
2t^2 - t - 3 = 0.
(2t - 3)(t + 1) = 0
t = 3/2 或者t = -1.
a/b - b/a - (a^2 + b^2)/(ab)
= -2b/a
= -2t
= -3或者2