You have the following quadratic equation:
x² - 9x - 8 = 0
In order to complete squares, it is necessary that the term 9x, must be two times x by a unknown coefficient b, that is,
9x = 2xb
by solving the previous equation for b you obtain:
9x = 2xb cancel out x factors both sides
9 = 2b divide by 2 both sides
9/2 = b
If you add b² = (9/2) = 81/4 you can complete the square of the polynomial, such as shown in the following:
x² - 9x + 81/4 - 8 - 81/4 = 0
The first three terms are a perfect square (x - 9/2)² , and the rest of terms result - 8 - 81/4 = (-32-81)/4 = -113/4.
Then, you have:
x² - 9x + 81/4 - 8 - 81/4 = 0
(x - 9/2)² - 113/4 = 0
Hence, the number b = 9/2 would have to be added to complete the square
ANSWER = 9/2