Not every optimization problem is a parabola. A firm's costs split in two, and the split is the whole idea. Fixed cost $F$ — rent, insurance, salaries — is owed in dollars each period whether or not a single unit is made. Variable cost $v$ is the dollars each unit costs to produce, and the selling price $p$ is the dollars each unit brings in. The difference $p - v$ is the contribution margin: what one unit contributes toward the fixed cost after paying for itself.

So the break-even output is just a division: $Q = \dfrac{F}{p - v}$ — Q equals F over p minus v, where $Q$ is the number of units that must be sold before the period turns a profit. Read the units as a guide: dollars divided by dollars-per-unit leaves units, which is exactly what Q should wear. If your rearrangement's units refuse to cancel that way it is wrong, no appeal — though units that DO work out never prove you right. The check is a one-way street.

One condition, and it is the difference between a business and a hobby: $p$ must exceed $v$. Sell a unit at or below what it costs to make and no quantity ever breaks even — every extra sale digs the hole deeper. And the myth this lesson exists to kill: a healthy margin per unit means NOTHING until the fixed cost is covered. Rent does not care how profitable each unit was.