Find the value of k = ( x^2 - 4 )/ ( 2x -5 ) if the roots of the equation are equal .

We'll multiply both sides by 2x -
5.

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

We'll simplify and we'll
get:

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

We'll remove the
brackets:

2kx - 5k = x^2 -
4

We'll move all terms to one
side:

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

We'll combine like
terms:

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

For the roots of the quadratic to be equal, the
discriminant delta has to be zero.

delta = b^2 -
4ac

a,b,c are the coefficients of the
quadratic.

delta = (-2k)^2 - 4(5k -
4)

delta = 4k^2 - 20k +
16

4k^2 - 20k + 16 = 0

We'll
divide by 4:

k^2 - 5k + 4 =
0

We'll apply quadratic
formula:

k1 = [5 + sqrt(25 -
16)]/2

k1 = (5+3)/2

k1 =
4

k2 = (5-3)/2

k2 =
1

The values of k, for the equation to have
equal roots, are: {1 ; 4}.