Al-Khwarizmi's major contribution to algebra   [Completing the Square]

     Al-Khwarizmi's  six equations

1. Squares equal to roots. e.g. x² = 10 x
2. Squares equal to numbers. e.g. x² = 39
3. Roots equal to numbers. e.g. 10 x = 39
4. Squares and roots equal to numbers; e.g. x² + 10 x = 39.
5. Squares and numbers equal to roots; e.g. x² + 21 = 10 x.
6. Roots and numbers equal to squares; e.g. 3 x + 4 = x².

The problem that Al-Khwarizmi  used:

... a square and 10 roots are equal to 39 units.

Express it mathematically

Answer:  x²+10x = 39 [note: Al-Khwarizmi expressed this in words, this is written in modern day expression for simplicity]

 The solution Al-Khwarizmi found:

             The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

Some Credits:

Sarton writes:-

... the greatest mathematician of the time, and if one takes all the circumstances into account, one of the greatest of all time....

, Gandz in [6] (see also [23]), argues for a very different view:-

Euclid's "Elements" in their spirit and letter are entirely unknown to [al-Khwarizmi]. Al-Khwarizmi has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind.

Conclusion

As can be seen al-Khwarizmi was a notable mathematician along with an abundance of other attributes. He discovered new ways of solving quadratic equations with algebra while keeping the problems simple and easy to manipulate. Al-Khwarizmi's ways of working with quadratic equations were so popular that his book Al-Jabr was used as the principle mathematics book at European universities until the 16th century (erols.com).

Click below for more information:

http://www-history.mcs.st-andrews.ac.uk/~history/Mathematicians/Al-Khwarizmi.html