2. The smallest integer $k$ that makes the quadratic equation in $x$
$$
2 x(k x-4)-x^{2}+6=0
$$

have no real roots is ( ).
(A) -1
(B) 2
(C) 3
(D) 4
2. B.

Organize the quadratic equation $2 x(k x-4)-x^{2}+6=0$, to get $(2 k-1) x^{2}-8 x+6=0$.
Since the equation has no real roots, then $\Delta=64-24(2 k-1) \leqslant 0$.
Solving for $k$ yields $k \geqslant 1 \frac{5}{6}$.
Therefore, the smallest integer $k$ is 2.