Answer
$k=1$
Work Step by Step
If the curve $y=x^2+k$ is tangent to the line $y=2x$, then the two curves intersect only at one point, the point of tangency. We equate the two expressions to determine what value of $k$ will create only a single point of intersection:
$$x^2+k = 2x$$
$$x^2-2x+k=0$$
$$(x-1)^2 + k-1=0$$
We only achieve one point of tangency at $x=1$, which is $k-1=0$ or $k=1$. Therefore $k=1$.
