tailieunhanh - Basic Mathematics for Economists - Rosser - Chapter 6

6 Quadratic equations Use factorization to solve quadratic equations with one unknown variable. Use the quadratic equation solution formula. Identify quadratic equations that cannot be solved. Set up and solve economic problems that involve quadratic functions. | 6 Quadratic equations Learning objectives After completing this chapter students should be able to Use factorization to solve quadratic equations with one unknown variable. Use the quadratic equation solution formula. Identify quadratic equations that cannot be solved. Set up and solve economic problems that involve quadratic functions. Construct a spreadsheet to plot quadratic and higher order polynomial functions. Solving quadratic equations A quadratic equation is one that can be written in the form ax2 bx c 0 where x is an unknown variable and a b and c are constant parameters with a 0. For example 6x2 7 0 A quadratic equation that includes terms in both x and x2 cannot be rearranged to get a single term in x so we cannot use the method used to solve linear equations. There are three possible methods one might try to use to solve for the unknown in a quadratic equation i by plotting a graph ii by factorization iii using the quadratic formula In the next three sections we shall see how each can be used to tackle the following question. If a monopoly can face the linear demand schedule p 85 - 2q 1 at what output will total revenue be 200 1993 2003 Mike Rosser It is not immediately obvious that this question involves a quadratic equation. We first need to use economic analysis to set up the mathematical problem to be solved. By definition we know that total revenue will be TR pq 2 So substituting the function for p from 1 into 2 we get TR 85 - 2q q 85q - 2q2 This is a quadratic function that cannot be solved as it stands. It just tells us the value of TR for any given output. What the question asks is at what value of q will this function be equal to 200 The mathematical problem is therefore to solve the quadratic equation 200 85q - 2q2 3 All three solution methods require like terms to be brought together on one side of the equality sign leaving a zero on the other side. It is also necessary to put the terms in the order given in the above definition of