Maths educator Po-Shen Loh has discovered a way to solve quadratic equations that is much more intuitive than $$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$, and is apparently unprecedented in the entire 4000-year history of thought on quadratic equations! See arxiv.org/abs/1910.06709.

The trick is as follows: Assume, first of all, that $$a=1$$, and let $$r$$ and $$s$$ be the (unknown) roots. Then we can write $$x^2 + bx + c = (x - r)(x - s)$$.

Expanding the right-hand side, we get $$x^2 + bx + c = x^2 - (r+s)x + rs$$.

So we have $$r + s = -b$$, and $$rs = c$$.

Now, the only way we can have $$r+s = -b$$ is if $$r = -\frac{b}{2} + z$$ and $$s = -\frac{b}{2} - z$$, for some $$z$$.

Since $$rs = c$$, this means that $$c = \frac{b^2}{4} - z^2$$.

Rearranging, we get $$z = \pm \sqrt{\frac{b^2}{4} - c}$$.

And now, substituting into the formula for $$r$$ and $$s$$, we get $$r, s = -\frac{b}{2} \pm \sqrt{\frac{b^2}{4} - c}$$.

But this is just the same as $$\frac{-b \pm \sqrt{b^2 - 4c}}{2}$$, which is exactly what we would expect from the standard formula, given that $$a=1$$! The general derivation, for $$a \neq 1$$, takes a few more steps, but is fairly straightforward.

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The innovation here seems to be in the emphasis placed on the properties of the sum of the roots and the product of the roots.

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