Non-Negative Integer Parameters for Integer Roots

For $x^2 - 5x + k = 0$ to have integer roots, the discriminant $D = 25 - 4k$ must be a perfect square. Since $k$ is a non-negative integer ($k \ge 0$):

• $k = 0 \implies D = 25 - 0 = 25 = 5^2$ (Square, roots are $0, 5$)
• $k = 4 \implies D = 25 - 16 = 9 = 3^2$ (Square, roots are $1, 4$)
• $k = 6 \implies D = 25 - 24 = 1 = 1^2$ (Square, roots are $2, 3$) For $k > 6$, $D < 0$ or non-square. Thus, there are 3 non-negative integer values of $k$ ($0, 4, 6$).