Find the sum of all values of $k$ for which $x^2 + kx - 9x + 25 - 9$ is the square of a binomial.

Answer 1

As in Solution 1, we factor $x^2 + kx - 9x + 25 - 9 = (x + a)^2$ for some number $a.$ Then
\[x^2 + (k - 9) x + 16 = x^2 + 2ax + a^2 = 0.\]We see that $a^2 = 16,$ so $a = \pm 4.$ Then $k = 9 \pm 8.$ The sum of all possible values of $k$ is $9 + 8 = \boxed{17}.$