$2 \cdot 127$ Equation $\sqrt{x+4}-\sqrt{x-3}+1=0$
(A) No roots.
(B) One real root.
(C) One real root and one imaginary root.
(D) Two imaginary roots.
(E) Two real roots.
(7th American High School Mathematics Examination, 1956)
[Solution] Given $\sqrt{x+4}+1=\sqrt{x-3}$, squaring both sides and simplifying yields
$$
8+2 \sqrt{x+4}=0 \text {, }
$$

this equation cannot hold.
Therefore, the answer is $(A)$.