Al-Khwarizmi and quadratic equations

The treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. In this book, which has given us the word 'algebra', al-Khwarizmi gives a complete solution to all possible types of quadratic equation. Here is al-Khwarizmi's solution of the equation
x2 + 21 = 10x.
What is most remarkable is that in this case he knows that the quadratic has two solutions:-

Halve the number of the roots. It is 5. Multiply this by itself and the product is 25. Subtract from this the 21 added to the square term and the remainder is 4. Extract its square root, 2, and subtract this from half the number of roots, 5. There remains 3. This is the root you wanted, whose square is 9. Alternatively, you may add the square root to half the number of roots and the sum is 7. This is then the root you wanted and the square is 49.

Now it is clear that al-Khwarizmi is intending to teach his readers general methods of solution and not just how to solve specific examples. This is clear from the was that he continues:-

When you meet an instance which refers you to this case, try its solution by addition, and if that does not work subtraction will. In this case, both addition and subtraction can be used, which will not serve in any other of the three cases where the number of roots is to be halved.

Know also that when, in a problem leading to this case, you have multiplied half the number of roots by itself, if the product is less than the number of dirhams added to the square term, then the case is impossible. On the other hand, if the product is equal to the dirhams themselves, then the root is half the number of roots.

JOC/EFR March 2006