To find the value of \(k\) for which \(gf(x) = 0\) has equal roots, we first find \(gf(x)\).

\(gf(x) = g(f(x)) = g(2x^2 - 3x) = 3(2x^2 - 3x) + k = 6x^2 - 9x + k\).

For the quadratic equation \(6x^2 - 9x + k = 0\) to have equal roots, the discriminant must be zero.

The discriminant \(b^2 - 4ac\) for the quadratic \(ax^2 + bx + c = 0\) is given by:

\(b^2 - 4ac = (-9)^2 - 4(6)(k) = 81 - 24k\).

Setting the discriminant to zero for equal roots:

\(81 - 24k = 0\).

Solving for \(k\):

\(24k = 81\)

\(k = \frac{81}{24} = \frac{27}{8}\).