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 al-Khwarizmi (Muhammad ibn Musa al-Khwarizmi who lived approximately between 780 and 850 AD in Baghdad). Al-Khwarizmi worked at the House of Wisdom in Baghdad and wrote a famous book Hisab al-jabr w'al-muqabala , 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 al-Khwarizmi in this book as follows:

What must be the square which, when increased by ten of its own roots, amounts to 39?

The equation can actually be described in modern-day symbols like this

Al-Khwarizmi solved this by adding 25, or 5 squared to both sides of the equation (he got 5 as 10/2) so that the left-hand side of the equation becomes a perfect square. In this case actually so does the right-hand 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


Al-Khwarizmi 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 Al-Khwarizmi.

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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