SOLUTION: Find the possible value(s) for k if the quadratic 3x^2 -2kx +4k=0 has two real roots

The quadratic  3x^2 - 2kx + 4k  has two real roots if and only is the discriminant 


    d = b^2 - 4ac is non-negative:  (-2k)^2 - 4*3*4k >= 0.


It inplies

    4k^2 - 48k >= 0

    k^2  - 12k >= 0

    k*(k - 12) >= 0


ANSWER.  EITHER k <=0  OR  k >= 12.


Note that when k = 0 or k = 12 (when d = 0), the two real roots merge into one real root of the multiplicity 2.