1. **Problem statement:** A movie theatre's profit $P$ depends on the number of $1 price increases $x$ as $$P = 20(15 - x)(11 + x).$$ We need to find: a) The vertex form of the profit equation. b) The ticket price that results in maximum profit, the maximum profit value, and the number of tickets sold at that price. 2. **Rewrite the profit equation:** Expand the expression: $$P = 20(15 - x)(11 + x) = 20[(15)(11) + 15x - 11x - x^2] = 20(165 + 4x - x^2).$$ Simplify inside the parentheses: $$P = 20(165 + 4x - x^2) = 20(-x^2 + 4x + 165).$$ Distribute 20: $$P = -20x^2 + 80x + 3300.$$ 3. **Convert to vertex form:** The quadratic is $$P = -20x^2 + 80x + 3300.$$ Factor out the coefficient of $x^2$ from the first two terms: $$P = -20(x^2 - 4x) + 3300.$$ Complete the square inside the parentheses: Take half of $-4$ which is $-2$, square it to get $4$. Add and subtract $4$ inside the parentheses: $$P = -20(x^2 - 4x + 4 - 4) + 3300 = -20[(x - 2)^2 - 4] + 3300.$$ Distribute $-20$: $$P = -20(x - 2)^2 + 80 + 3300 = -20(x - 2)^2 + 3380.$$ 4. **Interpret vertex form:** The vertex form is $$P = -20(x - 2)^2 + 3380,$$ where the vertex is at $x = 2$ and $P = 3380$. 5. **Find ticket price for maximum profit:** Since $x$ is the number of $1 price increases from the base price $11$, the ticket price at maximum profit is: $$11 + 2 = 13.$$ 6. **Maximum profit:** The maximum profit is $$P = 3380.$$ 7. **Number of tickets sold at maximum profit:** Tickets sold is given by $(15 - x) imes 20$ or from the original factors: Number of tickets = $20(15 - x) = 20(15 - 2) = 20 imes 13 = 260.$ **Final answers:** a) Vertex form: $$P = -20(x - 2)^2 + 3380.$$ b) Maximum profit is 3380 when ticket price is 13, and about 260 tickets are sold at this price.