We have k = (x^2 - 4)/( 2x - 5)

k = (x^2 - 4)/( 2x - 5)

=> x^2 - 4 = 2kx - 5k

=> x^2 - 2kx + 5k - 4 = 0

As the roots of the quadratic equation are equal, b^2 - 4ac = 0

=> (-2k)^2 - 4*( 5k - 4) = 0

=> 4k^2 - 20k + 16 = 0

=> k^2 - 5k + 4 = 0

=> k^2 - 4k - k + 4 = 0

=> k(k - 4) - 1(k - 4) = 0

=> (k-1)(k-4) = 0

=> k = 1 and k = 4

Therefore the values of k are 1 and 4.