The example in Exercise 1 demonstrates a Babylonian method for solving a quadratic equat...

The example in Exercise 1 demonstrates a Babylonian method for solving a quadratic equation. According to Professor Victor J. Katz in A History of Mathematics (New York: HarperCollins College Publishers, 1993), “whatever the ultimate origin of this method, a close reading of the wording of the [ancient clay] tablets seems to indicate that the scribe had in mind a geometric procedure [completingthe- square]. . . .”

One of the earliest explicit uses of the completing-thesquare technique to solve a quadratic is due to the ninth century Persian mathematician and astronomer Muhammed al-Khwa¯ rizmi¯. The following example demonstrates al- Khwa¯ rizmi¯’s use of completing the square to solve the equation x² + 8x = 84:

To find the positive root of

Begin with a square of side x, as in Figure A. Take half of the coefficient of x: this is one-half of 8, or 4. Now form two rectangles, each with dimensions 4 by x, and adjoin them to the square, as indicated in Figure B. Then, as in Figure C, draw the dashed lines to complete the (outer) square.

In Figure B, the combined area of the square and the two rectangles is x² + 8x. But from the equation we wish to solve, x² + 8x is equal to 84. In Figure C the area of the small square in the lower right-hand corner is 16. Thus the area of the entire outer square

is 16 + 84, or 100. But the area of the outer square is also (x _ 4)2. Therefore

(x + 4)² = 100 and, consequently, x + 4 = 10

The positive root we are looking for is therefore x = 6.

On your own or with a group of classmates, work through the preceding example, filling in the missing details as necessary. Then, on your own, use this completing-the-square process to find the positive root of each of the following equations.

Write out your solutions in detail, as if you were explaining the method to another student who had not seen it before. Be sure to include the appropriate geometric figures.