2.51 Given the equations $x^{2}+k x+6=0$ and $x^{2}-k x+6=0$, if the roots of the second equation are each 5 more than the corresponding roots of the first equation, then $k$ equals
(A) 5.
(B) -5.
(C) 7.
(D) -7.
(E) None of the above.
(14th American High School Mathematics Examination, 1963)
[Solution] Let the two roots of the first equation be $r, s$. By Vieta's formulas, we have
$$
r+s=-k, \quad r s=6 .
$$

For the second equation, we have
$$
\left\{\begin{array} { l } 
{ ( r + 5 ) + ( s + 5 ) = k , } \\
{ ( r + 5 ) ( s + 5 ) = 6 , }
\end{array} \text { or } \quad \left\{\begin{array}{l}
r+s=k-10, \\
r s+5(r+s)+25=6 .
\end{array}\right.\right.
$$

Substituting the first equation into the second, we can solve to get $k=5$.
Therefore, the answer is $(A)$.