Final answer:
To solve quadratic function word problems, translate the problem into a quadratic equation, use the Zero Product Property or the quadratic formula to find solutions, and interpret the solutions in context. Discard any solutions that don't make sense within the given scenario.
Step-by-step explanation:
To solve quadratic function word problems, it's essential to understand that these problems often involve a second-order polynomial, which is typically in the form of ax^2 + bx + c = 0. To tackle these problems, follow these steps:
- Read the problem carefully and identify the quantities involved and what you are being asked to find.
- Translate the word problem into a quadratic equation. This may involve setting up an equation based on the given information.
- If the quadratic is factorable, factor it and use the Zero Product Property to find the solutions. Otherwise, use the quadratic formula to find the roots of the function, where the roots are the solution for x in the equation ax^2 + bx + c = 0.
- Interpret the solutions in the context of the problem, and discard any that do not make sense (like a negative number for a length).
Here is an example:
Suppose a company's profit, P, in thousands of dollars, is given by the quadratic function P(x) = -2x^2 + 12x - 14, where x represents the number of items sold in hundreds. You are asked to find the number of items the company needs to sell to maximize its profit.
This is a maximum problem where the vertex of the parabola will give the maximum value. The x-coordinate of the vertex can be found using -b/(2a), which in this case is -12/(2*-2) = 3. The company needs to sell 300 items (as x represents hundreds) to maximize profit.