How do solve an Interesting Diophantine Equation

Question

I would like to know, for what integer values of $x$ makes $f(x)$ an integer for this equation, which I have derived from several other equations: $$f\left(x\right)=0.25x-0.5(33)+\left(\frac{1}{x}\cdot\frac{33^2}{4}\right)$$ It looks simple, but I would like a definitive answer. I'm leaving the fractions as decimals as I don't know if it would make much of a difference. As I'm studying efficient factorization techniques, I'm specifically interested in methods that do not include factoring integers (as it would be counter-productive to do so).

Answer 1

Write the equation as $$ y = \frac{x}{4} - \frac{33}{2} + \frac{1089}{4x} $$ For $y$ to be an integer, clearly $x$ must be a divisor of $1089 = 3^2 \cdot 11^2$; for each of the $18$ divisors (positive and negative), it turns out $y$ is an integer.