 # MSC # 21 – Using the Average: The Product of the Sum and Difference

I came across a YouTube video “A Different Way To Solve Quadratic Equations” at https://www.youtube.com/watch?v=ZBalWWHYFQc) published on Dec 12, 2019 by Po-Shen Loh, a Carnegie Mellon Math Professor and current USA Math Olympiad coach.

The “Po-Shen Loh” method as it came to be known is similar to our 21st Math Special for Christmas – Using the Average: The product of the Sum and Difference.

I was, of course, excited to know that someone with Prof Loh’s stature have just discovered in 2019 a method which I have been using for several years now and which I wrote about in my book “Algebra Made Easy as Arithmetic” which was published in January 2017. But while I used it only in factoring polynomials, Prof Loh used it in solving all types of quadratic equations including those with irrational or imaginary roots and even in deriving the quadratic formula.

I will discuss my “Using the Average” method, together with other Vedic Math techniques of solving quadratic equations on Dec 5, 2021in the Inspirational Maths from India (Year 4) webinars. We are inviting you to join me and some of the best VM practitioners in the world in these webinars. Just click on this think to register: https://bit.ly/IMI4Reg.

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 15th century mathematicians and the work of the Indians. He said that he found out that key points of this method have been discovered hundreds or thousands years ago and anyone can put it together.

(See how Prof Loh used it to solve all types of quadratic equations in “Examples: A Different Way to Solve Quadratic Equations” in https://www.youtube.com/watch?v=XKBX0r3J-9Y )

For my part, I got the idea for this method from two Math authorities. The first one was Dr. Arthur Benjamin. In his book The Secrets of Mental Math, he recalled that as a young boy he was thinking of the largest product that can be made from the numbers adding up to 20. He started in the middle 10 x 10 is 100, 11 x 9 is 99, 12 x 8 is 96, 13 x 7 is 91, etc.  He noticed that the products were getting smaller and their difference from 100 is 1, 4, 9, 16 … or 12, 22, 32, 42, etc.

Dr. Benjamin then tried numbers adding up to 26: 13 x 13 = 169, 14 x 12 = 168, 15 x 11 = 165 16 x 10 = 160, etc. And he noticed the products also became smaller and differ from 169 also by the same pattern 1, 4, 9, 16, etc.

Dr. Benjamin used the pattern he discovered to help him develop a very easy squaring technique.

Next, I saw this problem from the book Discovering Vedic Mathematics by Kenneth Williams of the Vedic Mathematics Academy: Solve x + y = 10 and xy = 24. Sir Ken, who is my VM mentor, wrote that the average of x and y is 5 and since the difference of the numbers from the average must be 52 – 24 = 1, the numbers must be 4 and 6.

I realized then that this is the same problem we encounter when we are factoring quadratics: the sum is the coefficient of the x or middle term while the product is the constant term. In factoring we try to to “guess” what two factors of the constant term would add up to the coefficient of the middle term.

Simply “using the average” will eliminate the guesswork.

And, instead of writing a new explanation for the technique, I will just quote what I have written in my book:

“With large values for B= 28 and C = 192, it may take some time if we were to use trial and error to find two factors of 192 which will add up to 28.

We shall apply the technique known as “using the average” to arrive at a faster solution. If the average of two numbers is a and if they differ from their average by d then one of the numbers is (a + d) and the other number is (a – d). Their product is (a2 – d2) which is 192 in this case.

Since 28 is the sum of the factors, 28/2 or 14 is their average and 142 = 196. Now d2 = 196 – 192 = 4 and therefore d = 2. The correct factors then are (14 + 2) or 16  and (14 – 2) or 12.

So x2 + 28x + 192 = (x + 16) (x + 12).