"The author would actually be very surprised if this approach has entirely eluded human discovery until the present day, given the 4,000 years of history on this topic, and the billions of people who have encountered the formula and its proof. Yet this technique is certainly not widely taught or known (the author could find no evidence of it in English sources)"
I certainly don't think it has eluded humany discovery until the present day. It's known to middle school students in Asia that multiplying the original equation so that A == 1 would greatly simplify the roots formula.
Comments
I have trouble understanding why this deserves a paper. This is what I learned in middle school back in Vietnam (although I did take advanced Math).
For equation Ax^2 + Bx^2 + C = 0, the roots are:
x1 = (-B + sqrt(B^2 - 4AC))/2A
x2 = (-B - sqrt(B^2 - 4AC))/2A
Now the author looks at a special case where A = 1. The equation becomes x^2 + Bx^2 + C = 0. Of course the roots simply become:
x1 = -B/2 + sqrt(B^2 - 4C)/2 = -B/2 + sqrt((B^2)/4 - C)
x2 = -B/2 - sqrt((B^2)/4 - C)
I certainly don't think it has eluded humany discovery until the present day. It's known to middle school students in Asia that multiplying the original equation so that A == 1 would greatly simplify the roots formula.
I think the same having attended a regular grammar school in Germany.
Yeah, half the elegance of the linked expressions just comes from setting A = 1.
This is definitely well-known. I learned this under the name of "Vieta's formulas" back in middle school. Vieta lived in the 1500s.
The a = 1 is not the important bit - that's probably nearly universal.