The quadratic formula is one of the top five formulas in math, and it is usually studied in high school, so it’s a good chance you’re familiar with it. By now, billions of people have had to learn it, memorize, and use the formula in order to solve quadratic equations.
However, according to mathematician Po-Shen Loh from Carnegie Mellon University, there’s actually been easier and better before, but it remained almost completely hidden for thousands of years. In a 2019 paper detailing the analysis, Loh says that the quadratic formula is a ‘remarkable triumph of early mathematicians’ dating back to the beginnings of the Old Babylonian Period around 2000 BCE, but also entirely acknowledges some of its ancient flaws.
“It is unfortunate that for billions of people worldwide, the quadratic formula is also their first (and perhaps only) experience of a rather complicated formula which they must memorize,” Loh writes in the paper.
A Genius Approach
That difficult task, carried out by about four millennia worth of math students, may not have been actually necessary, the genius says. Obviously, there have always been alternatives to this algebra formula, including factoring, completing the square, or even breaking out the graph paper.
However, the quadratic formula is typically considered as the most complete and dependable method for solving quadratic problems, even if it is a bit mysterious. This is what the quadratic formula looks like:
That formula can be employed to solve basic form quadratic problems, where ax2 + bx + c = 0. Back in September of 2019, Loh was thinking about the mathematics behind the quadratic equations when he came up with a new, simplified way of deriving the same precept. It was an alternative technique that he describes in his paper as a ‘computationally-efficient, natural, and easy-to-remember algorithm for solving general quadratic equations.’
“I was dumbfounded,” Loh says of the discovery. “How can it be that I’ve never seen this before, and I’ve never seen this in any textbook?”
In the mathematician’s new method, he starts from the basic principle of trying to factor the quadratic x² + bx + c as (x − )(x − ), which is similar to looking for two numbers to put the blanks with sum, b, and product c. He uses an average method that focuses on the sum, as opposed to the more typically taught way of concentrating on the product of two numbers that make up c, which needs guesswork to solve equations.
“The sum of two numbers is 2 when their average is 1,” Loh further explains on his website. “So, we can try to look for numbers that are 1 plus some amount, and 1 minus the same amount. All we need to do is to find if there exists a u such that 1 + u and 1 − u work as the two numbers, and u is allowed to be 0.”
As per Loh, a valid value for u can always be found when using his alternative quadratic method, in an intuitive way, making it feasible to solve any quadratic problem.
There’s no Clue Why This Escaped the Millennia
In his paper, Loh admits he would ‘be very surprised if this approach has entirely eluded human discovery until the present day, given the 4,000 years of history on this topic,’ but says the alternative way, which merges steps created by Babylonian, Greek and French mathematicians, is ‘certainly not widely taught or known (the author could find no evidence of it in English sources).’
Still, since first publishing his pre-print paper detailing the simple proof online in October, Loh says he noticed a 1989 research article that is the most similar previous research he has ever found. This work made Loh even more amazed that this alternative method has not been found before now.
“The other work overlapped in almost all calculations, with an apparent logical difference in assuming that every quadratic can be factored, and a pedagogical difference in choice of sign,” Loh explained.
All that there is to solve then is the enigma of why this method has not become more widely known before since it gives us ‘a delightful alternative approach for solving quadratic equations, which is practical for integration into all mainstream curricula,’ Loh said in his paper. This could also mean that nobody would need to memorize the quadratic formula again.
There’s still no clue as to why this escaped wider notice for millennia, but if Loh is correct, maths textbooks could be on the edge of a historic rewriting.
“I wanted to share it as widely as possible with the world,” Loh says, “because it can demystify a complicated part of maths that makes many people feel that maybe maths is not for them.”