Are quadratics that easy?

Algebra Level 3

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

(axb)2+(bxa)2=x ( ax-b)^2 + (bx-a)^2 = x

has at least one integer solution.

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


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