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Why can ALL quadratic equations be solved by the quadratic formula?
History of Quadratic Formula

How was the quadratic formula $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ found and proven?

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marked as duplicate by Rahul, Nate Eldredge, M Turgeon, Pedro Tamaroff, Matt N. Sep 3 '12 at 18:18

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Do you mean historically or how to prove it? –  vanguard2k Sep 3 '12 at 9:48
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Previously: History of quadratic formula (seems to be answered in the comments), and Why can ALL quadratic equations be solved by the quadratic formula? –  Rahul Sep 3 '12 at 9:51
    
ah, thanks for the edit. –  think123 Sep 3 '12 at 9:55
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For future reference, when you are adding tags to your question, please read the short descriptions that pop up and make sure they are appropriate. For example, the description for proof-theory says "Proof theory is an area of logic that studies proof as formal mathematical objects," which is not what your question is about. –  Rahul Sep 3 '12 at 10:04
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3 Answers

up vote 4 down vote accepted

It seems that Alkharazmi was the first person who found the formula in generla case. Though, he stated it in terms of Arabic words like "Xi", means "thing". How did it find it? Here is the summery of his work, in our language.

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I don't think that is true. al-Khwarizmi didn't have a concept of negative numbers, so he treated equations of the form $ax^2 + bx = c$, $ax^2 + c = bx$, and $ax^2 = bx + c$ as separate cases. –  MJD Sep 3 '12 at 11:16
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I don't know for sure, but I imagine it was first found by thinking about rearranging squares and rectangles, as in the following image:

enter image description here

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by completing the square. $a\neq 0$, $ax^2+bx+c=a\left(x^2+\dfrac{b}{a}x+\dfrac{c}{a}\right)=\cdots$

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This is way too sketchy. Perhaps a little more expanded development would have helped the OP better. –  DonAntonio Sep 3 '12 at 10:36
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I doubt that it's true that it was first discovered or proved by completing the square. –  Michael Hardy Sep 3 '12 at 12:48
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