Unexpectedly Intriguing!
20 December 2019

Sometimes, great insights come from tinkering. For many inventors, tinkering with older inventions is a time-tested method for gaining a better understanding of how they work and how they could work better. The insights they gain can then be applied in new ways, or perhaps combined with other insights in new ways, leading to new innovations.

Po-Shen Loh, a maths professor at Carnegie Mellon, has been tinkering with quadratic equations. In doing that, he has come up with a simpler proof of the quadratic formula. He explains the more intuitive approach he developed for finding the roots of quadratic equations in the following video:

As the video makes clear, the insights that Loh has put together in a new way have all been around for hundreds, if not thousands, of years. What's new and innovative is how Loh has combined them, which offers the promise of making it easier for students to understand how to solve quadratic equations, which ranks as one of the hardest things students learn in algebra classes.

If you didn't catch the 15 seconds near the end of the video with Loh's instructions for using his method for solving the quadratic equation, here they are:

### Alternative Method of Solving Quadratics

1. If you find r and s with sum -B and product C, then + Bx + C = (x - r)(x - s), and they are all the roots
2. Two numbers sum to -B when they are -(B/2) ± u
3. Their product is C when (/4) - = C
4. Square root always gives value u
5. Thus, -(B/2) ± u work as r and s, and are all the roots

If you're used to seeing quadratic equations of the form: ax² + bx + c = 0, we should note that Loh's method divides each term of this equation by a, where in his formulation, B = b/a and C = c/a. That leaves the term by itself, and of course, 0/a = 0, which all but eliminates the need to track the a in the traditional formulation any further.

On the other hand, if you have the equation in that traditional form, and you already know the quadratic formula, you can still use it - it hasn't changed and it still produces valid results. It may even be easier for given values.

If you want to go on a deeper dive into Loh's methods for solving quadratic equations, here's a 40-minute video with lots of examples and discussion of the insights involved.

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