A company's profit \( P \) in thousands of dollars can be modeled by the quadratic equation \( P(x) = -2x^2 + 12x - 16 \), where \( x \) is the number of units produced in hundreds. How many units should the company produce to maximize profit?

["Maximizing Profit: Optimizing Production Levels Using a Quadratic Model", "Understanding how to maximize profit is essential for any business aiming to grow sustainably. When dealing with quadratic profit models, identifying the optimal production level becomes crucial. This article explores how to determine the number of units a company should produce to achieve maximum profit from the given quadratic equation:\n[ P(x) = -2x^2 + 12x - 16 ]\nwhere ( P(x) ) is the profit in thousands of dollars, and ( x ) represents the number of units produced in hundreds.", "---", "The Nature of the Quadratic Profit Function", "The equation ( P(x) = -2x^2 + 12x - 16 ) is a downward-opening parabola due to the negative coefficient (( -2 )) of the squared term. This contrast bellows a key business principle: profit reaches a peak (maximum) at the vertex and declines afterward.", "Since the profit function is quadratic in nature, there is a single, unique point of maximum profit—the vertex—and this value lies precisely at the axis of symmetry of the parabola.", "---", "Finding the Vertex: Optimal Production Level", "For any quadratic equation in the form ( P(x) = ax^2 + bx + c ), the ( x )-coordinate of the vertex—where ( P(x) ) is maximized—is given by:\n[\nx = -\frac{b}{2a}\n]", "In our equation:\n- ( a = -2 )\n- ( b = 12 )\n- ( c = -16 ) (not needed for vertex calculation)", "Substituting into the formula:\n[\nx = -\frac{12}{2 \ imes (-2)} = -\frac{12}{-4} = 3\n]", "This result, ( x = 3 ), indicates that the company should produce 300 units (since ( x ) is in hundreds) to maximize profit.", "---", "Interpreting the Result: Practical Implications", "Producing 300 units positions the company at the peak of its profit curve. At this point:\n- The profit function reaches its highest value.\n- Any additional units beyond 300 will decrease total profit (since the parabola descends symmetrically).\n- Conversely, producing fewer than 300 units yields lower profits, despite reducing production costs.", "Understanding this balance enables data-driven decisions—avoiding underproduction (lost revenue potential) and overproduction (diminished returns).", "---", "Maximizing Real-World Profit: Key Takeaways", "To apply this model effectively:\n1. Replace ( x ) with actual hundreds produced (e.g., 3 = 300 units).\n2. Confirm the equation reflects real-world conditions: costs, pricing, and market demand.\n3. Use the result to guide operational planning—optimizing ordering, scheduling, and resource allocation.", "In essence, mathematics empowers strategic decision-making. By identifying ( x = 3 ), the company unlocks its profit-maximizing output, aligning efficiency with financial success.", "---", "Final Answer\nThe company should produce 300 units (when ( x = 3 )) to maximize profit under this model.", "---", "Keywords: profit maximization, quadratic profit model, business optimization, maximum profit x, calculate profit vertex, optimize production, expand profit function, analytics in business, high-profit strategy"]