Are quadratics that easy?

Algebra Level 5

Consider all pairs of non-zero integers \( (a,b) \) such that the equation

\[ ( ax-b)^2 + (bx-a)^2 = x \]

has at least one integer solution.

The sum of all (distinct) values of \(x\) which satisfy the above condition can be written as \( \frac{ m}{n} \), where \(m\) and \(n\) are coprime positive integers. What is the value of \(m + n \)?


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