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

The example in Exercise 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 [completing-the-square]….”

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

To find the positive root of x2 + 8x = 84:

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

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 complet- ing-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.

(a) x2 + 14x = 72

(b) x2 + 2Ax = B, where B > 0

Figure A

Figure B

Figure C

Exercise

More than 3000 years ago, the ancient Babylonian mathematicians solved quadratic equations. A method they used is demonstrated (using modern notation) in the following example.

To find the positive root of x2 + 8x = 84:

Rewrite the equation as x(x + 8) = 84, and let y = x + 8. Then the equation to be solved becomes xy = 84. Now take half of the coefficient of x in the original equation, which is 4, and define another variable t by t = x + 4. Then we have x = t — 4 and y = t + 4. Therefore

With t = 10 we get x = 10 − 4 = 6, the required positive root.

On your own or with a group of classmates, work through the above example, filling in any missing details if necessary. Then (strictly on your own), use the Babylonian method 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. This will involve a combination of English composition and algebra. Also, in part (b), check your answer by using the quadratic formula.

(a) x2 + 14x = 72

(b) x2 + 2Ax = B, where B > 0