To most of us, the words ‘quadratic’ and ‘equation’ might seem like a distant memory. As a refresher, a quadratic equation is any equation having the form ax² + bx + c = 0, meaning it contains at least one term that is squared.
Now if you’re a math pleb like me, you’ll have trouble even recalling whether this was ever brought up in the classroom at all back in the good old days of ‘O’ Level Elementary Math. And while it may seem a bit inconsequential to most of us, those still actively involved in the math industry might beg to differ.
So why am I revisiting mathematical PTSD now, at a time when my weapon of choice is words and not numbers? What if I told you that the pain-in-the-ass quadratic formula we were coerced to mindlessly learn by heart was actually all for nothing? That there existed a much easier, more intuitive way to go about solving these problems.
I wait in a corner at Starbucks, steeling myself for the skull-cracking night of Quadratic equations that await me.
Just last month, Poh-Shen Loh, an American-based mathematician at Carnegie Mellon University in Pittsburgh, born to Singaporean immigrants, wowed the industry by presenting an alternative and supposedly more intuitive working, angling it as A Different Way To Solve Quadratic Equations. He discovered this while analysing math curricula for schoolchildren, with the intention of developing new explanations that could help to make math less daunting and more approachable.
I was pressed to explore just how much this discovery impacted our society at large, and if it did at all. Seeing how I know next to nothing when it comes to arithmetic, I sat down with Bernard, a Computer Science major at Yale NUS, to probe further into this matter.
I wait in a corner at Starbucks, steeling myself for the skull-cracking night of Quadratic equations that await me. Moments later, Bernard walks in, and it doesn’t take him long to spot me. We exchange greetings and I leave to get him a drink. When I get back, I barely set his drink down before I notice that his stationery is already sprawled out—formula sheets and all.
He wastes no time in delving into the question on hand—what is this mystery method? As a refresher, he runs through the two most commonly used methods of solving the Quadratic equation with me, Completing the Square and the dreaded Quadratic Equation Formula we’ve all had to memorise.
“How do you even remember all this stuff?” I asked. Bernard briefly looks up and manages a tiny laugh. “I guess after revising for a while, it all just comes back to you”.
Bernard is patient and meticulous as he carefully explains Loh’s new found method step by step. He starts by pointing out that this new method is built on the fact that the quadratic equation formula can be split in the following way:
For those out there who are as lost as I am, we are looking to solve for x. The algebraic expressions B, C, R and S are placeholders that will be substituted with real numbers to aid us in getting our answer.
In a swift rearrangement, we can then conclude that the right-hand side equals 0 for both occasions when x = R and when x = S.
When this is true, then R x S = C, and R + S = -B. For simplicity’s sake, Bernard uses the example of x² + -7x + 12 to illustrate. Referring to Loh’s method above, this would mean that R = 3, and S = 4.
Using R x S = C, we can conclude that C = 12, and through R + S = -B, we can derive that -B = 7.
Now comes the bit that’s a little harder to grasp, yet is the part that makes all the difference. Bernard highlights that the average of R and S surely has to be ±B/2.
So, moving forward, it is paramount that we use ±B/2, to form ±B/2 ± z, where z is a single unknown quantity. We do this in order to derive the value of ±x.
Quoting Loh himself, “using that, we can now multiply these numbers together to get an expression for C”. From there, we can easily solve the rest of the equation, making it an efficient method compared to its older counterpart
If that still didn’t cut it for you, here’s a video where Loh explains his new method more succinctly.
Bernard effortlessly breezes through the explanation of the method, as if he’s been practicing it over the past couple of years. And here I am, clutching my seat, thinking back to when I first laid eyes on the article, and struggling to catch anything at all from it. Despite the great job he did with the demonstration, it still takes me a considerable while to even begin to understand how the new method works.
“This new method is simply a re-discovery, or a re-angling.”
When asked for his opinion on how ground-breaking he finds this method, Bernard simply doesn’t find it to be mind-blowing or revolutionary because he would categorise this as more of a re-framing of the equation. “If you do some rearrangements, you’ll find that you actually arrive at the same formula that’s being used now,” he concludes.
He explains that rather than a discovery of a new method to solve the Quadratic equation, this new method is simply a re-discovery or a re-angling of how to tackle a Quadratic equation. “In fact,” he adds, “there are many more ways to solve the Quadratic equation, just that we don’t use them in pedagogy or in this part of the world”.
Even after all the ramble and learning, I still have no idea how quadratic equations are used in my daily life. In the life of a writer, where math is left in the dust, it’s just difficult to feel the connection to claims of such revolutionary methods. Off the top of his head, Bernard cited calculating profit, speeds, and determining scores in athletics as some examples of quadratic equations used in daily life.
There’s a good chance you’ll spend four seconds skimming through the article of Poh Shen Loh’s new method and move on blissfully with life.
However, it’s not like accountants and auditors are out here scribbling formulas on stacks of papers to complete their day-to-day tasks. Instead, all of these processes are mostly, if not completely automated. Be it on Microsoft Excel, Tableau, SQL or other data mining softwares, most solutions are just a click of a button away.
Simply put, no one actually uses the quadratic equation in its long-form, but rather, the equation is often utilised as a tool or framework to derive the answers they need. And this tool is oftentimes coded into an algorithm, and then forgotten. So, it seems like the only people to whom the quadratic equation method still matters, are educators.
Unless you’re teaching a math class in secondary school or doing something along similar lines, there’s a good chance you’ll spend four seconds skimming through the article of Poh Shen Loh’s new method and move on blissfully with life.
Moving away from the quadratic equation, Bernard draws my attention to a concept that we can draw similarities from—the Pythagoras Theorem.
“I actually just so happened to read up on the history of the Pythagoras Theorem, and realised that just like Poh Shen Loh’s method, the theorem is also merely a simplification of a method that’s much longer and more complicated”, Bernard explains.
He proceeds to demonstrate how the formula we know all too well—c²= a² + b²—was initially derived from a humble square. Many rounds of simplification later, I’m beginning to gain clarity on the concept and how it really is a mere reduction of the many other methods that can also be used to solve for the longest side of a triangle. Helping to put things into perspective, Bernard finally succeeds in helping me wrap my head around the crux of the whole matter.
The short answer is, Loh’s method isn’t that much of a deal, where innovation is concerned. Bernard is pessimistic that the method will be adopted into our education system. After all, it is only human of us to be resistant to change. We’re so set in our ways that it could be years or even decades before methods start to shift.
As the saying goes, if it ain’t broke, don’t fix it. And that’s the sentiment that both of us share with regards to Loh’s new method. It just seems like something too trivial to fuss over, and simply too inconsequential for most of us to even spend more than five minutes of our attention on.
Taking a step back from all these, I start thinking about how many other ‘quadratic equation’ situations we might find ourselves in, in life. I am spiralling down the rabbit hole of “how many other things have I been doing ‘wrong’ in life this whole time? And how many more lie in wait of discovery?”
Bernard chimes in with words of wisdom: “Maybe we’re not supposed to have all the answers”. And it’s true, I guess a big part of the human condition is knowing what we don’t know, and even more so, not knowing what we don’t know. Because if we did, then what fun would life be?
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