Chapter 11-SOLVING EQUATIONS IN EXCEL USING GOAL SEEK To solve a single algebraic equation with Goal Seek, proceed as follows: 1. Enter an initial guess in one of the cells on the worksheet. 2. Enter a formula for the equation, in the form f(x) = 0, in another cell. Within this formula, express the unknown quantity x as the cell address containing the initial guess. 3. From the Ribbon’s Data tab, click on the What-If Analysis button within the Data Tools group. Then select Goal Seek from the resulting drop-down menu. 4. When the Goal Seek dialog box appears, enter the following information: (a) The address of the cell containing the formula in the Set cell entry location. (b) The value 0 in the To value entry location. (c) The address of the cell containing the initial value in the entry location labeled By changing cell. Then select OK.

Example 11.5 Solving a Polynomial Equation in Excel Using Goal Seek In Example 11.2, we found that the equation 𝑓(𝑥) = 2 𝑥 5 − 3 𝑥 2 − 5 = 0 has a real root at approximately x = 1.4. We will now use Excel’s Goal Seek feature to obtain a more accurate solution. Let us select the value x = 1.4 as a first guess, since we already know that this is the approximate value of the root.

Example 11.6 Convergence Considerations Sometimes a great deal can be learned by studying a problem whose solution is well known. Thus, let us use Goal Seek to determine the roots of the equation tan⁡(𝑥) = 0 within the interval 0 ≤ x ≤ π. Anyone who has ever taken a course in trigonometry should recognize that this equation has roots at x = 0 and x = π, and a discontinuity at x = π/2.

Problem 11.17 Determine the smallest positive root and the largest negative root (the negative root closest to the origin) for the equation: 𝑥 tan⁡𝑥 = 2

SOLVING EQUATIONS IN EXCEL USING SOLVER Excel includes another feature, called Solver, which may be able to obtain a converged solution when Goal Seek cannot. Solver is entirely independent of Goal Seek. To solve a single equation with Solver, proceed as follows: 1. Enter an initial guess in one of the cells on the worksheet. 2. Enter a formula for the equation, in the form f(x) = 0, in another cell. Within this formula, express the unknown quantity x as the cell address that contains the initial guess. 3. From the Ribbon’s Data tab, click on Solver within the Analysis group. The Solver Parameters dialog box will then appear. Like Goal Seek, convergence is not guaranteed. Hence, Solver will generate an error message if the computation does not converge, or if a mathematical error (e.g., an attempt to calculate the square root of a negative number) is detected during the course of the computation. Solver will also generate an error message if it detects a circular reference in the formula for f(x).

Example 11.7 Solving a Polynomial Equation in Excel Using Solver In several previous examples we solved the polynomial equation 𝑓(𝑥) = 2 𝑥 5 − 3 𝑥 2 − 5 = 0 and found a positive real root in the vicinity of x = 1.4. Let us now use Solver to find a root of this equation, with the added restriction that x ≥ 0. Use a convergence criterion of 10-5 (i.e., 0.00001).

Problem 11.19 Many consumers buy expensive items, such as cars and houses, by borrowing the purchase cost from a bank and then repaying the loan on a constant monthly basis. If P is the total amount of money that is borrowed initially, the amount of the monthly payment, A, can be determined from the formula 𝐴=𝑃 𝑖 (1+𝑖) 𝑛 (1+𝑖) 𝑛 −1 where i is the monthly interest rate, expressed as a fraction (not a percentage), and n is the total number of payments. Suppose you borrow $10,000 to buy a car. If you are required to repay $350 each month for 36 months, what is the corresponding monthly interest rate?