Finding the Sum of A + \sqrt{A^2 - B^2} and U + \sqrt{U^2 - V^2}

[Bruno Tolentino]

Given two numbers: $$A + \sqrt{A^2 - B^2}$$ and $$U + \sqrt{U^2 - V^2}$$ OBS: A, B, U and V are real numbers.

I want sum it and express the result in the same form: $$A + \sqrt{A^2 - B^2} + U + \sqrt{U^2 - V^2} = x + \sqrt{x^2 - y^2}$$ So, x depends of A and U. And y depends of B and V:
$$x = x(A, U)$$ $$y = y(B, V)$$ Do have any ideia about how do it?

PS: A is the arithmetic mean of the roots of the quadratic equation and B is the geometric mean. Is a nice expression for the quadratic formula!

 

[mfb]

There is no unique solution. You can simply set x=y= [left side of the equation], for example. And I don't see any special solution sticking out.