From the first equation, p-1/p=q^2. From the second equation, p+1/p=4/q. Summing both equations, 2p=(4+q^3)/q. Subtracting both equations, 2/p=(4-q^3)/q. Multiplying both results, 4=(16-q^6)/q^2, so q^6-4q^2-16=0. Writing q^2=r we get r^3-4r-16=0 which has a single real root r=2, so if q=√2 then p=1+√2, and if q=-√2, then p=1-√2.