How was the quadratic formula found and proven?

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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?

Previously: History of quadratic formula (seems to be answered in the comments), and Why can ALL quadratic equations be solved by the quadratic formula? — , Sep 03 '12 at 09:51
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score 8 · Answer 1 · answered Sep 03 '12 at 17:29

8

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:

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answered Sep 03 '12 at 17:29

Nick Alger

18,844

score 5 · Accepted Answer · answered Sep 03 '12 at 10:13

5

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.

answered Sep 03 '12 at 10:13

Hooman

3,647

1

http://en.wikipedia.org/wiki/Quadratic_equation#History and http://mathworld.wolfram.com/QuadraticEquation.html – lab bhattacharjee Sep 03 '12 at 10:18
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 03 '12 at 11:16

score -1 · Answer 3 · edited Sep 03 '12 at 12:43

-1

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

edited Sep 03 '12 at 12:43

Michael Hardy

1

answered Sep 03 '12 at 09:51

h kone

11

1

This is way too sketchy. Perhaps a little more expanded development would have helped the OP better. – DonAntonio Sep 03 '12 at 10:36
1

I doubt that it's true that it was first discovered or proved by completing the square. – Michael Hardy Sep 03 '12 at 12:48

How was the quadratic formula found and proven?

3 Answers3