x^2 + 3x - 9 = 0
There's a 3 on the x in the middle. Half of 3 is 3/2. 3/2 squared is 9/4.
x^2 + 3x + 9/4 - 9/4 - 9 = 0
(x + 3/2)^2 - 9/4 - 9 = 0
(x + 3/2)^2 - 9/4 - 36/4 = 0
(x + 3/2)^2 - 45/4 = 0
(x + 3/2)^2 = 45/4
x + 3/2 = +- sqrt(45/4)
x + 3/2 = +- 3sqrt(5)/2
x = +- 3sqrt(5)/2 - 3/2
x = (+- 3sqrt(5) - 3)/2
x = 3(+- sqrt(5) - 1)/2
x = 3(-sqrt(5) - 1)/2 or x = 3(sqrt(5) - 1)/2