Possible Integer Values of k for Quadratic Equations

For $2x^2 + kx + 5 = 0$ to have no real roots, its discriminant must be negative: $D_1 = k^2 - 4(2)(5) < 0 \implies k^2 < 40 \implies -\sqrt{40} < k < \sqrt{40}$ Since $k$ is an integer, $k \in \{-6, -5, \dots, 5, 6\}$.

For $x^2 + (k - 5)x + 1 = 0$ to have two distinct real roots, its discriminant must be positive: $D_2 = (k - 5)^2 - 4(1)(1) > 0 \implies (k - 5)^2 > 4$ $\implies k - 5 > 2 \text{ or } k - 5 < -2 \implies k > 7 \text{ or } k < 3$

Combining both conditions for integer $k$: $k \in \{-6, -5, -4, -3, -2, -1, 0, 1, 2\}$ There are 9 such integer values for $k$.