中四數學問題(quadratic equation)

As the roots are rational nos, the discriminant of this equation is a perfect square.
△ = 7² - 4*3*k = m^2, where m is an integer, say.
==> 12k = (7 + m)(7 - m)
==> (7+m, 7-m) = (12, k), (2k, 6), (3k, 4), (4k, 3), (6k, 2), (12k, 1)
==> (m, k) = (5, 2), (1, 4), (3, 10/3), (4, 11/4), (5, 2), (6, 13/12)
(as k > 0, so only consider k is positive)

Therefore, the possible values of k is 2, 4, 10/3, 11/4, 13/12.

However, this is only valid for m is an integer, if m is a rational no,
the possible solutions for k is more than this, ie, k = 9/16.
So, this question miss some condition(s), ie. if the question is :

Write down all possible positive integers of k such that ...

Then the answer is k = 2 or 4.

2013-10-21 10:55:28 補充：
If the said condition is added, then the larger value of k is 4, the equation is
3x^2 + 7x + 4 = 0
==> (3x + 4)(x + 1) = 0
==> x = -4/3 or -1

since 3x^2+7x+k=0 has rational roots,
7^2-4(3)(k)>=0 (b^2-4ac>=0)
49-12k>=0
49>=12k
k<=49/12
from the given that 0<k<=49/12
the larger value of k is 49/12
3x^2+7x+49/12=0
36x^2+84x+49=0
(6x+7)^2=0
x= -7/6

OK, Thank you very much.

2013-10-21 02:34:24 補充：
but when k = 49/12, the roots (-7/3) is a rational number, why it needs to be a square?

2013-10-21 02:37:27 補充：
OK , I get it , It avoids something like root 2 from the discriminant

2013-10-21 02:40:31 補充：
Good night . Masterijk

Rational roots mean the roots are rational numbers.

You need the discriminant being a perfect square.

△ = 7² - 4*3*k = 49 - 12k ≥ 0

12k ≤ 49

k ≤ 49/12

k is positive, then 0 < k ≤ 49/12.

2013-10-21 02:24:56 補充：
If k is limited to be an integer, then k can only take 1, 2, 3, 4.
k = 1　＝＞　△ = 37
k = 2　＝＞　△ = 25 is a perfect square
k = 3　＝＞　△ = 13
k = 4　＝＞　△ = 1 is a perfect square

2013-10-21 04:00:00 補充：
郭老師，其實今次你不必刪答呀～

你答對了，是版大問得不清楚～

有可能同學是手誤打漏了一些資料～

你答的是正解～

即使是 integer，你也可以補充～

下次不要浪費你的心血～

你答得很好～　可以拿最佳的～

以示尊重，我不會答此題！ ^___^

大家要努力！