The quotient of nine and a number, increased by half of the number

We can write in math as 9/x + x/2.  We can simplify by cross-multiplying to get 18=x^2.

Now we find the square root of 18, which is simplified to 3sqrt2. Because squaring a number will always leave you with a positive number, whether the original number was positive or negative, both positive 3sqrt2 and negative 3sqrt2 are answers.