decide what number must be added to the binomial to make a perfect square x^2+(3/2)x

The general form of a 2nd degree polynomial that is a perfect square is Ax^{2} + 2ACx + C^{2}.

In this case, 2AC = 3/2. Since A = 1,

2C = 3/2 and C = 3/4

So the number that needs to be added is C^{2} = (3/4)^{2} = 9/16

and our polynomial is x^{2}+ (3/2)x + 9/16