Suppose positive integers p,q,r are such that 

pq = 32 and q = k²/2

Then 

p(k²/2) = 32   

pk² = 64

p = 64/k²

There are 4 square numbers k² which are factors of 64.

They are 1, 4, 16, and 64.

Since q must be half a square number, we can eliminate odd number 1.

So q can only be 2, 8, or 32

Therefore there are three solutions:

1.  2 is a factor of 32 and also 2 is half of 4 which is a square number, 22.

2.  8 is a factor of 32 and also 8 is half of 16 which is a square number, 42.

3.  32 is a factor of 32 and also 32 is half of 64 which is a square number, 82.

Edwin