Three thousand years BC, an official under ancient Babylon came to the farmer and said that the wheat tax for this crop would increase.
When one of them wanted to expand both the length and width of his field by the same distance x, the farmer encountered an equation of the form:Â Ax2Â + Bx + C = 0. That may have been the first time that
Throughout history, problems requiring humans to solve quadratic equations have appeared in every civilization, from Babylon, Egypt, India to China.
Thanks to applications from basic to great, quadratic equations are today included in every high school mathematics curriculum in the world.
This is really hard to remember and not intuitive at all.
That’s why Po-Shen Loh, a mathematician at Carnegie Mellon University and coach of the US Olympic math team, wanted to find a simpler, more intuitive and deductive solution to the step equation.
Po-Shen Loh, coach of the US Olympic math team
In 2019, Po-Shen Loh published a scientific article sharing his new method of solving quadratic equations.
Let’s find out what method Po-Shen Loh used:
1. Suppose, we have the following quadratic equation: x2 + Bx + C = 0
2. With a little observation or recalling Viete’s theorem, we can see that any polynomial on the left side can be factored into the form:
If the left side is 0, this equation will have a solution of x =R or x = S. Basically, those are the two solutions of the given quadratic equation.
Now, when multiplying the right side to break the parenthesis, we will have:
This equation is equivalent to -B = R+ S and C = R.S.
3. Po-Shen Loh found that if -B is the sum of R and S, then the average of R and S will be -B/2.
Then we simply switch sides and take the square root to find z:
4. Insert z back into R and S, we will get 2 solutions of the original equation:
TADA!
Suppose we have a function y = x2 – 4x -5.
According to Viete’s theorem, R+S = 4, the average of R and S is 2.
z is half the distance between R and S, being an unknown quantity, then R =2-z and S = 2+z.
Applying Viete’s theorem further, we have R.S = -5.
By breaking the brackets, we get 4-z2=-5
Equivalent to z2=9, z=3.
So the solution to the first equation is R =2-3= -1 and S = 2+3= 5.
Continuing to try Po-Shen Loh’s solution with another quadratic equation with complex solutions, we see that it is still correct.
Once we are familiar with Po-Shen Loh’s method, we can quickly memorize this equation and it will have 2 solutions: -B/2 ± z.
So, finally the two solutions of the original equation are:
This same equation, if solved using the solution formula, will become much more complicated.
If the equation is of the form Ax2 + Bx + C = 0, simply divide all the coefficients by A to get a new equation of the form x2 + (B/A) x + C/A = 0 and apply the solution of
In 2019, as soon as Po-Shen Loh’s scientific paper was released on arXiv.org, a series of math teachers were surprised by this solution.
The question is why until now no one has accidentally discovered this method and shared it widely?
However, he said that so far no one has ever combined these two solutions together, to teach students a simple but very logical way of thinking when finding solutions to quadratic equations.
In the article, Po-Shen Loh said he had found all the documents recording the method of solving quadratic equations of ancient Babylonians, Chinese, Greeks, Indians and Arabs as well as mathematicians.
The results showed that none of them had ever solved a quadratic equation the way he did, even though Viete’s theorem and its Babylonian expansions existed hundreds or thousands of years ago.
Maybe it’s because of our approach to quadratic equations, says Po-Shen Loh.
In addition, quadratic equations are now only associated with exercises in textbooks.
Unlike in the past, the Babylonians used quadratic equations in practical problems they encountered in life.
Therefore, finding new solutions is emphasized, and to do that, our ancestors in the past had more creative methods.
Now, after discovering a new way to solve quadratic equations, Po-Shen Loh has applied it right into his teaching programs.
But the best thing for Po-Shen Loh is probably the students’ enthusiastic reception of this new solution.
Po-Shen Loh said this solution also emphasizes a philosophy in his teaching method.
Once students find math interesting and understand it, they will no longer be afraid of math or afraid of math.
After all, mathematics after thousands of years is still very alive and attractive.
References MIT, Nytimes