# 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 $$p+q$$.

×