A new way to make quadratic equations easy

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)

> "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.

We want to write x^2 + bx + c = 0 in the form (x+m)^2 + n = 0, so there's only one x left and we can rearrange for it.

Expanding, (x+m)^2 = x^2 + 2mx + m^2, so we get the x^2 we want, and the coefficients of x tell us b = 2m, so m = b/2. We also get an m^2 (= b^2/4) we don't want, so let's take it away:

(x + b/2)^2 - b^2/4 = x^2 + bx

That x + b/2 is x - (-b/2), x minus the average of the roots, which is the x value the parabola is centred on. Then we add c:

(x + b/2)^2 - b^2/4 + c = x^2 + bx + c

To find the roots, set it to 0 and rearrange for the one x that's left:

(x + b/2)^2 - b^2/4 + c = 0

(x + b/2)^2 = b^2/4 - c

x + b/2 = ±√(b^2/4 - c)

x = -b/2 ± √(b^2/4 - c)

Note that this is the average of the roots ± the article's z. Then combine:

x = -b/2 ± √((b^2-4c)/4)

x = -b/2 ± √(b^2-4c)/2

x = (-b ± √(b^2-4c))/2

If you have ax^2+bx+c = 0, divide the equation by a first, then do the same steps and you get the normal quadratic formula:

x = (-b ± √(b^2-4ac))/(2a)

I think the linked post misstates the purpose of the article: it's not about new maths, but about pedagogy and ways of explaining the quadratic formula.

> [...]

> 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.