Find the value of k for which 4x^3+kx-27 = 0 has two equal roots

Answer 1

4(x-a)(x-b)^2 = 4(x^3 + mx - 27/4) must have no x^2 term. (k=4m)
Also, ab^2 = 27/4 = 3^3 so
a = 3 and b^2 = 9/4
a quick check shows that b = -3/2 works.
(x-3)(2x+3)^2 = 4x^3 - 27 x - 27
so k = -27