Cost Optimization Given Constraints

Question

Balance = 1,360 USD

Cost = (0.0001% * Purchase Qty * Purchase Qty) + (1% * Purchase Qty)

Problem: Find maximum Purchase Qty while keeping Cost below Balance.

Is there a closed form solution to this?

Answer 1

Let the quantity be $q$. $$1360=10^{-6} \cdot q^2$$ $$q^2=1.36 \cdot 10^9$$ $$q = \sqrt{1.36 \cdot 10^9} \approx 3.688 \cdot 10^4$$