Hi Jamie,
It's generally going to be easiest to complete the square if the coefficient in front of the x^2 (or in this case, q^2) term is 1, so start by multiplying through by 2. This gives:
q^2 - 5/2q -6 = 0.
Then, take the coefficient of the linear term (-5/2), divide it by 2, add it to q, and square the quantity. This gives:
(q-5/4)^2
Expanding this out would give q^2 - 5/2q + 25/16. Notice that the only difference between this and your equation is that we want -6 but we have 25/16. So we simply subtract to get what we want, or:
(q-5/4)^2 - 25/16 - 6 = 0
(q - 5/4)^2 = 6 + 25/16
(q- 5/4)^2 = 121/16
If you needed to solve for q, you would simply take a square root and get:
q - 5/4 = +/- 11/4
q = 16/4 = 4,
q = -6/4 = -3/2