Quadratic Equation: Who Invented & What's the Proof?

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Discussion Overview

The discussion centers on the origins of the quadratic equation and the various proofs associated with it. Participants explore different methods of deriving the quadratic formula, including completing the square and alternative approaches presented in educational literature.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Meta-discussion

Main Points Raised

  • One participant inquires about the inventor of the quadratic equation and the proof behind it.
  • Another participant suggests looking at external resources for proof, specifically mentioning MathWorld and Wikipedia.
  • Some participants assert that the proof involves completing the square, although the exact historical attribution remains uncertain.
  • A participant references a different derivation method from the latest edition of the Mathematics Teacher, describing it as elegant but does not provide details initially.
  • Later, a participant summarizes an article by Henry Piccioto, outlining a method that involves the roots of the quadratic function and the coordinates of the vertex, emphasizing a visual approach rather than the traditional completing the square method.
  • There are questions about the new method and expressions of curiosity regarding its details.
  • Another participant explains the notation "^" as exponentiation, indicating a side discussion on mathematical symbols.

Areas of Agreement / Disagreement

Participants express curiosity about the proofs and methods, but there is no consensus on the historical attribution of the quadratic equation or the superiority of one proof method over another. Multiple competing views on the derivation methods remain present.

Contextual Notes

The discussion includes references to external sources and specific articles, which may contain additional assumptions or methods not fully explored in the thread. The historical context of the quadratic equation's invention is not definitively established.

Who May Find This Useful

Readers interested in the history of mathematics, proofs of mathematical concepts, and alternative methods of teaching quadratic equations may find this discussion relevant.

thharrimw
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who came up with the Quadratic equation and what is the proof behind it?
 
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It's just completing the square, but used on the general form of the quadratic equation. I can't remember who first proved it though :( Some Arabic mathematician i think.
 
Take ax^2+bx+c=0 complete the square and you'll obtain the quadratic equation.
 
The latest edition of the Mathematics Teacher, published by the NCTM has a different way to derive the quadratic formula than completing the square.

It is actually kind of elegant as well.
 
Quantumduck said:
The latest edition of the Mathematics Teacher, published by the NCTM has a different way to derive the quadratic formula than completing the square.

It is actually kind of elegant as well.

what is it?
 
thharrimw said:
what is it?

Yeah I'm curious too
 
Dang, I knew I should have brought my copy home. I will look it up and post later.
-------
Ok, looked it up. It is complex, so I will have to short cut it. The full citation, if interested is:
Piccioto, Henry. (February 2008). A new path to the quadratic formula. Mathematics Teacher 101:6, 473-478.

If there are roots p and q, then the function can be written in factored form
y=a(x-p)(x-q) = x^2 - a(p+q)x + apq

It follows that the product of the roots is c/a, since c=apq and the sum of the roots is -b/a, since b= -a(p+q).

From here, he uses that information to find (h,v), the co-ordinates of the vertex. The average of the roots, h, is -b/2a. This is then substituted into the formula to get v, and the resultant is

v= (-b^2 +4ac)/4a

Notice that this is the discriminant divided by 4a!

Finally, the author notes that the x intercept is on either side of the vertex by the same amount, d, so x = -b/2a +- d, and if we move the parabola so that the vertex is at the origin, it's equation simply becomes y=ax^2.

With this new translated parabola, we can then do a little algebra (which is explained in the article, 2 steps) to get x = the negative boy couldn't decide on whether to attend a radical party or be square, so he missed out on 4 awesome chicks and the party was all over by 2 am.

It is a very visual method, instead of the normal completing the square method.

There is no way I did it justice in my re-telling.
 
Last edited:
what is ^?
 
  • #10
Ephratah7 said:
what is ^?

Exponentiation. It means "raised to the power of"
 

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