two inequalities

Explanation as follows:

圖片參考：http://i238.photobucket.com/albums/ff245/chocolate328154/Maths373.jpg?t=1218449762


2008-08-11 18:21:45 補充：
Please note the difference between [non - negative] and [positive]:
For non - negative, 0 is included and so in Q1 △= 0 is taken in to account, which makes △<= 0
For positive, 0 is NOT included and so in Q2 △ < 0.

2008-08-12 23:36:01 補充：
http://hk.knowledge.yahoo.com/question/question?qid=7008062100946
For your 1st question, you may read the explanation in the above link.

Then for the 2nd one, I have mentioned that before, but maybe not clearly enough, so let me clarify: [ non - negative ] means >= 0 while [ positive ] means > 0

1.
For the expression to be non-negative, delta must be smaller than 0, that is
(k-4)^2 - 4(k^2 - 5k + 4) < 0
k^2 + 16 - 8k - 4k^2 + 20k - 16 < 0
-3k^2 + 12k < 0
-k^2 + 4k < 0
k(k - 4) > 0
So 0 > k > 4.
2.
Same as Q1. delta < 0. Therefore,
4(k-3)^2 -8(k+1) < 0
4k^2 + 36 - 24k - 8k - 8 < 0
4k^2 - 32k + 28 < 0
k^2 - 8k + 7 < 0
(k - 7)(k -1) < 0
So 1< k < 7.