Explanation

Put A=sqrt(p+x), B=sqrt(p-x). Then (A+B)/(A-B)=p; componendo-dividendo gives A/B=(p+1)/(p-1). Squaring: (p+x)/(p-x)=(p+1)^2/(p-1)^2. Solving, x[(p-1)^2+(p+1)^2]=p[(p+1)^2-(p-1)^2], i.e. 2x(p^2+1)=4p^2, so x=2p^2/(p^2+1). Blind squaring without componendo-dividendo is messier but equivalent.