2.287. For what values of $a$ and $b$ does the quadratic trinomial $16 x^{2}+144 x+(a+b)$ represent a perfect square, given that $b-a=-7$?
Solution.

The trinomial $16 x^{2}+144 x+(a+b)$ represents a perfect square when $b-a=-7$, if the discriminant is zero, i.e.

$$
\left\{\begin{array} { l } 
{ b - a = - 7 , } \\
{ 1 4 4 ^ { 2 } - 4 \cdot 1 6 ( a + b ) = 0 , }
\end{array} \left\{\begin{array}{l}
b-a=-7, \\
b+a=324
\end{array} \Rightarrow a=165.5 ; b=158.5\right.\right.
$$

Answer: $a=165.5 ; b=158.5$.