# “Using The Average” method of solving quadratic equations

Last Dec 21,
in our 21^{st} MATH-Inic Special for Christmas, I discussed about what the
very popular YouTuber Presh Talwalkar of MindYourDecision referred to as “a
different method of solving quadratic equations that everybody is talking about”.

This further strengthened my resolve to promote Vedic Mathematics in the Philippines because I have been using that method after reading the “Using the Average” chapter of Kenneth Williams book, “Discover Vedic Mathematics”

I have also written about this method in my book, “Algebra Made Easy as Arithmetic” which was first published in Feb 2017. (Please see the accompanying photo taken from page 82.)

The exact example Sir Ken gave was: Solve x + y = 10, xy = 24. Sir Ken, who is the founder of the Vedic Mathematics Academy(UK) wrote that the average of x and y must be 5 and since their product is 24, their difference from the average must be 5^{2} – 24 = 1. The numbers must therefore be 5 ± 1 = 6, 4.

The problem
struck me as exactly the same problem we face regularly when we are trying to
solve quadratic equations by factoring. Using
the same figures, if we are to solve x^{2} + 10x + 24 = 0, we will list
down the factor pairs which will give a product of 24 and determine which pair
will have a sum of 10.

I also recalled reading Dr Arthur Benjamin’s “Secrets of Mental Math” where he recounted that during his early teen years (when he was still unfamiliar with Algebra), he was thinking of the largest product which can be made from the numbers which will sum up to 20. He first tried splitting 20 in the middle (which is the average) into 10 and 10 and got 100. Then 11 x 9 = 99, 12 x 8 = 96, 13 x 7 = 91, etc. He found out that the product becomes smaller and their difference from 100 are 1, 4, 9, etc which are the squares of 1,2, 3, etc.

Next, he tried getting the products of numbers adding up to 26. Starting from 13 x 13 = 169, then 14 x 12 = 168, 15 x 11 = 164 and so on. He found out that products behaved exactly as before – differing from the maximum value 169 by by 1, 4, 9 or by the squares of 1,2, 3 etc.

We also know that for a given perimeter, the largest area that can be covered is a square.

But while, Dr. Benjamin used his discovery to develop his own squaring method, we thought that we can avoid the trial and error method factoring we were trained to do since high school and directly solve for the correct factors if we were to use the technique suggested by Sir Ken’s problem

I was pleasantly surprised that several days before Dec 21, I came across a Youtube video “A Different Way to Solve Quadratic Equations” posted by Po-Shen Loh, a Carnegie Mellon Math professor and US International Math Olympiad coach in which he discussed a method he “accidentally discovered” was like the “using the average” technique I wrote about.

Prof. Loh recounted that he accidentally discovered a method to
solve quadratic equations in a really simple way. “I was dumbfounded!”, he
said, “How can it be that I have never seen this before and I have never seen
it in any textbook.” Then he studied the works of the Babylonians, of 15^{th} century
mathematicians and the work of the Indians. He said that he found out that key
points of this method has been discovered hundreds or thousands years ago and
anyone can put it together.

I used it only in factoring but Dr Loh showed that it can be used to solve all types of quadratic equations and also to derive the quadratic formula. In another video, Dr Loh said that the key points for his method came from Viete’s Theorem and techniques used by ancient Babylonians. “Hopefully” he said, “by combining these insights from ancient Mathematicians, all of our future generations will not have to do any guessing.”

After Dr Loh posted a video in YouTube on Dec 12, 2019, many others followed including Presh Talwalkar because they thought that this method will become very popular in the academic world.

I just thought that if some older Mathematicians have read Ken William’s “Discover Vedic Mathematics” when it was first published in June 1984, when Dr Loh was only 2 years old, this generation could have long avoided the “guess and check” system of factoring.

I will discuss more examples of applying the “using the average” method in my future posts.