$$\left(\frac {p^2} {2(p^2+1)}\right)^2+\left(\frac p {2(p^2+1)}\right)^2=\frac p {p^2+1}\cdot \frac {p^2} {p^2+1}\Leftrightarrow p^4+p^2=4p^3\Rightarrow p=2+\sqrt 3 \vee p=2-\sqrt 3$$