What is the sum of all possible values of $k$ for which $x^2 + kx - 12x + 16$ is the square of a binomial?
x^2 + kx - 12x + 16 we can write
x^2 + ( k - 12)x + 16
If this can be expressed as the square of a binomial....it will have only one root
Thus....the discriminant will = 0 ....so.....
(k - 12)^2 - 4(16) = 0
(k - 12)^2 - 64 = 0
(k - 12)^2 = 64 taking both roots, we have that
k - 12 = ±√64 so either
k - 12 = 8 or k - 12 = -8
add 12 to both sides
k = 20 or k = 4
Check
x^2 + 8x + 16 factors as ( x + 4)^2
And
x^2 - 8x + 16 factors as (x - 4)^2