p and q are two numbers each greater than zero. √(p^2 + 5q) = 8 and √(p^2 – 3q) = 6. Find the values of p and q.

First of all, we have to raise to the power of two the first equation and will obtain: p^2 + 5q = 64. 
We have to proceed the same for the second equation and will obtain: p^2 - 3q = 36. 
Second step is to substract the equations we just got and will have: p^2 + 5q - p^2 +3q =  28, hence 8q = 28, so q = 28/8 = 7/2. We go back to the first or the second equation and plug in q and we obtain p^2 = 64-35/2, so p^2 = 93/2, so q = sqrt(93/2). So, p = sqrt(93/2) and q = 7/2. The solution is verified by the two equations and is available as both numbers are positive, as required.