Writing It Up · Hard problem

To prove $2^n \ge n^2$ for $n \ge 4$, the inductive step writes $2^{k+1} = 2 \cdot 2^k \ge 2k^2 \ge (k + 1)^2$. A careful write-up must say for which $k$ the last inequality holds. What is the smallest integer $k$ for which $2k^2 \ge (k + 1)^2$?