For which value of k will the roots of the equation 2x^2-5x+k=0 be real and rational numbers

The trick here is to find the discriminant, b^2-4ac, of the equation 2x^2 - 5x + k.
Here a=2, b=-5, and c=k.

So long as the discriminant is 0 or greater than 0, your roots will be real and rational numbers.

Thus, compute b^2-4ac:  (-5)^2 - 4(2)(k)

Let this equal zero and solve for k:  25=8k; k=25/8.

So long as k is 25/8 or smaller, b^2-4ac will be 0 or greater, and the roots will be real and rational.