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I am referring to this question about calculating roots of quadratic equations using average of roots instead of sum.

The author has claimed that his method is more effecient than the method that we generally use.

I am unable to understand how he can say this when the ultimate formula which he drives is same and involves same number of arithmetic operations.

Abstract This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the potential to demystify quadratic equations for students worldwide"

Or I have not understood his point well ?

Matthew Towers
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gpuguy
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    As I understand it, the point is that the derivation avoids the "completing the squares" step which students find mystifying. I have not seen the claim that the final formula is somehow more efficient...as you remark, it's just the usual formula. – lulu Jul 05 '20 at 14:47
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    I read that as "not unnecessarily complicated", not "more efficient" than anything. This is a pretty subjective question, since nobody has defined "efficient". – aschepler Jul 05 '20 at 14:48
  • In the abstract as you can see the author has clearly mentioned 'efficient', 'computationally light'. Of course this is in comparison to usual method. That is why I am wondering how is this efficient and 'computationally light'. But I should have contacted the author first before posting here. So will do it now. – gpuguy Jul 06 '20 at 16:34

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Of course the final formula according to Loh's method should be the same as the usual quadratic formula because otherwise it would be wrong. But Loh's point is not to memorize a formula and simply plugin the coefficients. Rather, it's easier to find the answers through an algorithm. You can arrive at the same answer (roots of the quadratic equation) through different algorithms. Some algorithms are more efficient than others. Consider the two simple formulas, $ab+ac$ and $a(b+c)$. You get the same final answer either way. But the former has two multiplications and one addition whereas the latter has only one multiplication and one addition. So the latter is computationally more efficient.

For finding the roots of a quadratic equation, completing the square is one algorithm and Loh's method is another one. Given $x^2+bx+c=0$, Loh's method is to find the midpoint $m$ between the roots ($m = -b/2$) and the distance $d$ between the midpoint and a root ($d^2=m^2-c$). Notice that $d^2$ is calculated using $m$. When you have $m$ and $d$, you can finally get the roots: $x = m \pm d$. Loh's method is considerably efficient when $b$ is an even integer because then you don't have to deal with fractions unless $c$ is also fraction.

Franko Zargoza
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According to this video, the quadratic formula on average is faster algorithmically.

I think the Poh Shen Loh method (which is really just quadratic formula in disguise, nevertheless) is probably better for humans but poor for computers.

https://www.youtube.com/watch?v=eZ5U0f14LaY&t=281s

user29418
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