Using the Average method of Solving Quadratic Equations III

Dr. Po-Shen Loh, in his YouTube video, “A different way to solve quadratic equations”( https://www.youtube.com/watch?v=ZBalWWHYFQc)  mentioned about studying the works of the ancient Babylonians, 15th- century Mathematicians and even works of Indians. He found out that the key points of this “new” method were discovered hundreds or thousands of years ago and anyone can put it together.

The algorithm presented here is from an article in Wikipedia (see “Wikipedia contributors. (2020, January 17). Quadratic equation. In Wikipedia, The Free Encyclopedia. Retrieved 01:27, January 19, 2020, from https://en.wikipedia.org/w/index.php?title=Quadratic_equation&oldid=936170451”). It is probably one of the key points referred to by Prof. Loh.

It shows the step by step method of solving the simultaneous equations x + y = p and xy = q. This is equivalent to finding the roots x and y  in the equation z² + q = pz. This can be converted into z² – pz + q =0 and can also be expressed as (z – x) ( z- y) = 0.

We will use Dr. Poh’s first example to understand the Babylonian algorithm: x² – 8x + 12 = 0. Here, p = 8 and q = 12.

1. Compute half of p : 8/2 = 4
2. Square the result: (4)² = 16
3. Subtract q: 16 – 12 = 4
4. Find the positive square root of 4: √4 = 2
5. Add the results of steps ( 1) and (4) to get x: x = 4 + 2 = 6       

y = p – x = 8 – 6 = 2

In another video “Examples: A different way to solve quadratic equations” (https://www.youtube.com/watch?v=XKBX0r3J-9Y) Dr. Loh explained that we usually solve quadratic equations by the factoring method and using his exact figures, to solve the equation x² – 14x + 24 = 0

                        x² – 14x + 24 = ( x – _ )(x – _ )

                        IF can find 2 numbers with

                                    Sum = 14 and

                                    Product = 24

                        THEN those are all the solutions.

The method we are taught to use since our high school days is to

1. find the factors pairs of 24: 24 x 1; 12 x 2; 8 x 3; and 6 x 4
2. “guess” which pair will add up to 14 and
3. “check” if the pair will really add up to 14.

But if we start with the sum,

1. we know that the average of the two roots are 14/2 or 7
2. one must be a certain number u below 7 or (7 – u) and the other must be u units above 7 or (7 + u)
3. The product of (7 + u) and (7 – u) which is 7² – u² is equal to 24.
4. u² is thus equal to 49 – 24 = 25 and u = 5
5. Therefore 7 + u = 12 and 7 – u = 2

So x² – 14x + 24 = ( x – 12 )(x -2 ).

This is the third of five parts of the article “Using the Average method of solving Quadratic equations”. The first two parts are: https://www.math-inic.com/blog/using-the-average-method-of-solving-quadratic-equations/ and https://www.math-inic.com/blog/using-the-average-method-of-solving-quadratic-equations-ii/

To be continued: