Completing the Square 

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General quadratic equations of the form were known to Egyptian, Babylonian, Chinese, Indian and Greek mathematicians. Arab mathematicians made progress on the solutions to these equations, in particular the famous alKhwarizmi (Muhammad ibn Musa alKhwarizmi who lived approximately between 780 and 850 AD in Baghdad). AlKhwarizmi worked at the House of Wisdom in Baghdad and wrote a famous book Hisab aljabr w'almuqabala , from which we derive the meaning and the name of algebra. One of the famous problems which introduces the 'completion of the square' was described by alKhwarizmi in this book as follows:
The equation can actually be described in modernday symbols like this AlKhwarizmi solved this by adding 25, or 5 squared to both sides of the equation (he got 5 as 10/2) so that the lefthand side of the equation becomes a perfect square. In this case actually so does the righthand side, but that is not always the case, nor is it necessary. So we get This can now be easily solved by taking the square root of both sides or AlKhwarizmi did not take into account the negative solution (that come some centuries later) so the only solution to this equation is x = 3. This is the diagram which represents the problem (but you can see the interactive version, if you have Geometer's Sketchpad, here). See how completing the square works in general  learn how to do it.

See how completing the square works in general  learn how to do it.
If you have geometer's sketchpad, download a file here.
See more about AlKhwarizmi. See a page on the development of algebra. See other pages on AS Level Maths, or download resources for AS Level. 

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