1. Let $n$, $b$, and $k$ be integers, and $\left(k-\frac{b}{2}\right)^{2} \leqslant n$. The equation $x^{2}+b x-k^{2}+k b+n=0$ has real roots. Consider the following four conclusions:
(1) The equation must have two distinct real roots;
(2) The equation has exactly one integer root;
(3) $b$ is divisible by 2;
(4) $n$ is a perfect square or zero. Which of the following conclusions are correct? ( ).
(A)(1)(2)(3)(4)
(B)(2)(3)(4)
(C)(2)(3)
(D) (1)
-1 . (B).
Since the equation has real roots, its discriminant $\Delta \geqslant 0$, i.e., $\left(k-\frac{b}{2}\right)^{2}=n$. From the given, we have $\left(k-\frac{b}{2}\right)^{2}=n$, which implies that $b$ is an integer multiple of 2. By $\Delta=0$, we know that $n$ is a perfect square or zero, and the equation has a unique integer root $x=\frac{b}{2}$.