The question is: Find all real values k such that the circle $x^2+y^2=9$ and parabola $y=x^2+k$ have no points of intersection.
It is clear that when $k > 3$, no points of intersection exist. There is also another range when $k$ is negative, so when you substitute:
$y-k = x^2$
$y-k+y^2 = 9$
$y^2 + y - (k+9) = 0$
The discriminant must be negative so
$1 + 4(k+9) < 0$
$k < -37/4$
However, the original range $k > 3$ disappeared. Does anyone know what I did wrong?