Question

18．若n为整数，且$\sqrt{{n}^{2}+9n+30}$是自然数，则n=-14或-7或-2或5．

Answer 1

分析 设$\sqrt{{n}^{2}+9n+30}$=p，再把等式两边同时乘以4，利用平方差公式把等式左边化为两个因式积的形式，列出关于p、n的方程组，求出n的值即可．

解答 解：∵设$\sqrt{{n}^{2}+9n+30}$=p（P为非负整数），则n²+9n+30=p²，
∴4n²+36n+120=4p²，
∴（2n+9）²+39=4p²，
∴（2p+2n+9）（2p-2n-9）=39，
∴$\left\{\begin{array}{l}2p+2n+9=1\\ 2p-2n-9=39\end{array}\right.$或$\left\{\begin{array}{l}2p+2n+9=39\\ 2p-2n-9=1\end{array}\right.$或$\left\{\begin{array}{l}2p+2n+9=3\\ 2p-2n-9=13\end{array}\right.$或$\left\{\begin{array}{l}2p+2n+9=13\\ 2p-2n-9=3\end{array}\right.$，
解得$\left\{\begin{array}{l}p=10\\ n=-14\end{array}\right.$或$\left\{\begin{array}{l}p=10\\ n=5\end{array}\right.$或$\left\{\begin{array}{l}p=4\\ n=-7\end{array}\right.$或$\left\{\begin{array}{l}p=4\\ n=-2\end{array}\right.$，
∴n=-14或-7或-2或5．
故答案为：-14或-7或-2或5．

点评 本题考查的是二次根式的性质与化简，先根据题意把原式化为两个因式积的形式是解答此题的关键．