Happy poly-new year!

Solve the quadratic equation and we will get

$x = 7 \pm \sqrt{49 + k}$

For the roots to be integers, $49 + k$ must be a perfect square but must not exceed $2063$

So, the possibilities of $49 + k$ are $7^{2}, 8^{2}, 9^{2}, ..., 45^{2}$

So there are $39$ possibilities for $k$