For grass, Nitrogen is usually the major requirement. The first number in the fertilizer composition is the % of nitrogen. The liquid fertilizers usually weigh about 10 lbs per gallon. 10-10-10 would contain 1 lb of nitrogen per gallon. Most places, 1 lb of Nitrogen per 1,000 square feet is recommended. 1/2 acre would be roughly 22,000 square feet so 22 gallons would be needed. 22 gallons @ $8 per gallon = $176.00
Check with a local farm supply or feed supplier. Pasture growers usually buy the liquid in bulk. Around here, 30-0-0 sells for about 20 cents per lb. Since it's 3 times as concentrated only about 7gallons would be needed. This stuff weighs 11 lbs per gallon, 7 gallons @ 11 lbs = 77 lbs @.20 per lb = $15.40
Depending on your grass type and weather, you can see the difference in a week. Be ready to mow! Also, you can add weed killer and take care of the weeds with the same application. Weedkillers like 24D are inexpensive.
Check with your local Ag Extension for recommended application rates in your area.