All of us remember quadratic equations, probably not fondly. If the need arises to solve one in our professional lives, our usual solution is to use the quadratic equation, which we may or may not have memorized through having had it drilled into our heads in Algebra I. The quadratic equation is what you get from applying the “complete the square” method to \(ax^{2}+bx+c\). Unless you’re a mathematician, that probably doesn’t mean much to you, even if you understood it at the time.
The mathematician Po-Shen Loh has discovered an incredibly simple method of solving quadratic equations that, astoundingly, doesn’t seem to have been known before. It’s based on the observation that if \(r\) and \(s\) are the roots of \(x^2+bx+c\) then \[x^2+bx+c = (x-r)(x-s) = x^2 -(r+s)x +rs\] so the sum of the roots is equal to \(-b\) and their product is \(c\). Therefore, the problem reduces to finding two numbers whose sum is \(-b\) and product is \(c\).
That may not seem a whole lot easier but Loh observed that if \(r+s=-b\) then their average is \(-b/2\). Two numbers will sum to \(-b\) precisely when their average is \(-b/2\). Those two numbers must be of the form \((-b/2-u)\) and \((-b/2+u)\) (because the average of two numbers lies mid way between them) so we need only find \(u\) such that \(b/2-u\) and \(-b/2+u\) have the product \(c\). That yields the equation \(b^{2}/4 -u^2 =c\). That’s easy to solve for \(u\) but the important point is there’s nothing to remember except that the roots sum to \(-b\) and multiply to \(c\).
You can find more details from Loh’s paper on the method, which is easy to read, but if you want to see a very simple demonstration of the method complete with examples take a look at his video. Watch the video and try a couple of examples to set the method and you’ll never have to worry about the quadratic equation again.