# Using the Average method of Solving quadratic equations IV

I wrote to Dr. Po-Shen Loh that I previously used a method similar to his “new approach”, and in response, he gave me an address where we can chat.

Dr Po-Shen Loh sent me the following links: https://www.poshenloh.com/quadraticrelated, where he collected verifiable dated references he came across in researching his new approach and https://arxiv.org/pdf/1910.06709.pdf , his formal article on this novel method. He also pointed to two articles in Spanish, the contents of which, he briefly explained to me.

For those who can not access the articles above at this time, this slide from one of his YouTube videos concisely summarized his approach.

The first step has been know hundreds of years ago and this is found in most high school algebra textbooks as Viete’s formula.

The next three steps were used by the Babylonians and the Greeks thousands of years ago and they were the subject of my previous post.

Step 5 is Dr. Loh’s derivation after combining the key points in the early methods.

When I told Dr. Loh about my previous post on the Babylonian method, he made an interesting comment: “perhaps the reason why Babylonians thought of that method of solution was because they couldn’t memorize the multiplication table. They used base 60, and so their multiplication table would be 60×60 instead of our 10×10 multiplication table. Instead, some scholars believe that Babylonians often used the difference-of-squares trick to multiply numbers. For example, to calculate 17×21, it is possible to just calculate (19-2)(19+2) = 19^2 – 2^2. Babylonians had tables of perfect squares, which they could use to subtract from. If using that perspective to think about multiplication, it becomes very natural to come up with the Babylonian technique of solving the problem of finding two numbers with given sum and product!”

To be continued: