2.286. For what value of $k$ can the polynomial $x^{2}+2(k-9) x+$ $+\left(k^{2}+3 k+4\right)$ be represented as a perfect square?
Solution.

The polynomial $x^{2}+2(k-9) x+\left(k^{2}+3 k+4\right)$ can be represented as a perfect square if the discriminant is zero, i.e., $(2(k-9))^{2}-4\left(k^{2}+3 k+4\right)=0, \quad 4\left(k^{2}-18 k+81-k^{2}-3 k-4\right)=0$, $-21 k+77=0$, from which $k=\frac{77}{21}=\frac{11}{3}$.

Answer: $\frac{11}{3}$.