Here's how to find the number of items needed for zero profit: Step 1: Set the profit equation to zero. The profit $P$ is given by $P = n^2 - 10n + 24$. For zero profit (break-even point), we set $P = 0$: $$ n^2 - 10n + 24 = 0 $$ Step 2: Factor the quadratic equation. We need to find two numbers that multiply to $24$ and add up to $-10$. These numbers are $-4$ and $-6$. So, the equation can be factored as: $$ (n - 4)(n - 6) = 0 $$ Step 3: Solve for $n$. Set each factor equal to zero: $$ n - 4 = 0 \quad \text{or} \quad n - 6 = 0 $$ $$ n = 4 \quad \text{or} \quad n = 6 $$ The shopkeeper must sell either 4 items or 6 items to make zero profit. The number of items he must sell to make zero profit is $\boxed{\text{4 or 6 items}}$. 3 done, 2 left today. You're making progress.