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

Last Wednesday evening, I came across a Youtube video “A different Way To Solve Quadratic Equations” at https://www.youtube.com/watch?v=ZBalWWHYFQc) published just last Dec 12, 2019 by Po-Shen Loh, a Carnegie Mellon Math Professor and current ISA Math Olympiad coach. I found out later in Wikipedia that “under his coaching, the United States International Math Olympiad team won in 2015, 2016, 2018 and 2019 – their first victories since 1994”.

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.

It turned out that
the “new” method he was referring to is exactly the same topic of our 21st MATH-Inic Special for
Christmas, which was scheduled to be posted today Dec 21, 20. This is “Using the Average: The product of the
Sum and Difference”. In fact, we
have already prepared an infographic for finding the binomial factors of x^{2}
+ 28x + 187, as an example, as early as Wednesday noon.

I was, of course, excited to know that someone with Prof Loh’s stature have just discovered a method which I have been using for several years now. 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 derive the quadratic formula.

(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 )

Last Thursday, I
realized that using 187 as the constant term is not a good example because it
can only be factored in two ways, 187 x 1 and 17 x 11. So I asked our graphic artist
to make another infographic, this time using x^{2} + 28x + 192 because
192 has many factors.

Last night when I started to write this post, I reviewed Po-Shen Loh’s video and I noticed that the “up Next” video was by Presh Talwalkar of MindYourDecision. It was sub-titled “A different way to solve quadratic equations that everyone is talking about.”

After enumerating Dr. Loh’s credentials, including CalTech-BS Math; Cambridge-MS Math; Princeton-PhD Math, Presh added “If Professor Loh is suggesting a problem solving method, It is definitely worth trying.”

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 1^{2},
2^{2}, 3^{2}, 4^{2}, 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 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 5^{2} – 24 = 1, the numbers must be 4 and 6.

Now 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.

I was a little worried that I may be accused of just copying from Prof. Loh, given the timing that he announced his discovery just a few days ago. I tried to recall the time I first used this method so I browsed through the past issues of our shelved “There’s a Math Teacher in the House Newsletter” and I saw that I wrote about it in the March 14, 2017 issue.

Now it is my turn to be dumbfounded. Not only was the example in the newsletter taken from my book “Algebra Made Easy as Arithmetic” p 82 but it was exactly the same example that we used in the infographic: x^{2} + 28x + 192!

Nevertheless, there are more than 4000 math lovers who own copies of my “Algebra …” could attest that I have used that method earlier than Jan 2017.

And, instead of
writing a new explanation for the technique, I will just quote from the 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 (a^{2} – d^{2}) which is
192 in this case.

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

So x^{2}
+ 28x + 192 = (x + 16) (x + 12)

More example can be found in Chapter 21 of our forthcoming book, “30 Master Strategies in Computing.”