Expected Difference Surds

Calculus Level 5

Let \(m,n\) be two randomly chosen real numbers satisfying that the equation \[\sqrt{x-m}+m=\sqrt{x-n}+n\] has a single real solution \(x=x_1\).

If \(\sqrt{x_1-m}+m=\sqrt{x_1-n}+n=P\), then the expected value of \[P-x_1\] can be expressed as \(\dfrac{p}{q}\) for positive coprime integers \(p,q\). Find \(p+q\).

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