To find the value of \(k\), we equate the curve and the line:

\((2k - 3)x^2 - kx - (k - 2) = 3x - 4\)

Rearrange to form a quadratic equation:

\((2k - 3)x^2 - (k + 3)x - (k - 6) = 0\)

For the line to be tangent to the curve, the discriminant of this quadratic must be zero:

\(b^2 - 4ac = 0\)

Here, \(a = 2k - 3\), \(b = -(k + 3)\), \(c = -(k - 6)\).

Calculate the discriminant:

\((k + 3)^2 + 4(2k - 3)(k - 6) = 0\)

Simplify:

\(k^2 + 6k + 9 + 8k^2 -60k + 72 = 0\)

\(9k^2 - 54k +81 = 0\)

Divide by 9:

\(k^2 - 6k + 9 = 0\)

\(k = 3\)

Since the line is tangent, \(k = 3\).