inequalities

Q1.
For the quadratic function (m-1)x^2 - 3x +(m -1) > 0.
there are 2 conditions to fulfill:
1) Delta < 0 and
2) The coefficient of x^2 term must be positive, so that only a minimum can exist, not a maximum. That is (m-1) > 0, or m> 1.
Therefore, m< -1/2 is rejected.
Q2.
For a quadratic equation to have 2 unequal roots, delta > 0. Therefore,
(k+1)^2 - (4)(8)(k-5) > 0
k^2 + 1 + 2k - 32k + 160 > 0
k^2 - 30k + 161 > 0
(k- 23)(k - 7) > 0
Therefore, k> 23 and k < 7.

2008-08-24 16:56:22 補充：
For further condition that the 2 roots must be positive, that means (b/a) and (c/a) in the standard quadratic equation must be positive. That means (k+1)/8 > 0 and (k-5)/8 > 0. So combining the results, 5< k < 7 and k > 23.

2008-08-24 17:03:33 補充：
Correction: Should be (-b/a), not (b/a). Sorry.

As follows:

圖片參考：http://hk.geocities.com/stevieg_1023/0AMTH3.gif


2008-08-24 18:14:49 補充：
SORRY FOR TYPING MISTAKE

THE ANSWER IS 5<7 OR k>23