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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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2$\begingroup$ Do you mean historically or how to prove it? $\endgroup$– vanguard2kSep 3, 2012 at 9:48
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2$\begingroup$ Previously: History of quadratic formula (seems to be answered in the comments), and Why can ALL quadratic equations be solved by the quadratic formula? $\endgroup$– user856Sep 3, 2012 at 9:51
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$\begingroup$ ah, thanks for the edit. $\endgroup$– LucasSep 3, 2012 at 9:55
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6$\begingroup$ 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. $\endgroup$– user856Sep 3, 2012 at 10:04
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3 Answers
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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:
answered Sep 3, 2012 at 17:29
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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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1$\begingroup$ en.wikipedia.org/wiki/Quadratic_equation#History and mathworld.wolfram.com/QuadraticEquation.html $\endgroup$ Sep 3, 2012 at 10:18
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$\begingroup$ 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. $\endgroup$– MJDSep 3, 2012 at 11:16
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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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1$\begingroup$ This is way too sketchy. Perhaps a little more expanded development would have helped the OP better. $\endgroup$ Sep 3, 2012 at 10:36
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1$\begingroup$ I doubt that it's true that it was first discovered or proved by completing the square. $\endgroup$ Sep 3, 2012 at 12:48