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The Quadratic equation formula of Al Khwarizmi

Why, when the original proof (which I shall reveal to you) is simple and beautiful, are schoolchildren expected to take an important piece of mathematics on trust? I mean, the formula enabling the solution of equations in x² (x-squared) of the form:


For me, and perhaps for you also, up until the famous quadratic equation formula, school algebra was logical and transparent, even if at times trickier than Latin. Then suddenly, like a bolt from heaven, came a horrible-looking formula, the solution to the above equation, that we were supposed to take on faith alone:


Do not misunderstand me: I would not criticise the teaching I received many years ago at school. Our teachers were dedicated to teaching, and loved their subjects. In those days also they had the freedom to teach a few things that were off the syllabus. We lived in a time when professionals were respected a little more than they are today, and were trusted to know their job and how to do it. Thus, as students we were not subjected to time-wasting and soul-destroying examinations merely for the sake of allowing semi-educated politicians to entertain the illusion that they could measure the qualities of experts (end of rant).

Even so, our teachers were under some pressure of time, because of examinations, and to the best of my memory this is why we were told to memorise the quadratic equation formula instead of being led gently through the proof.

Enough preamble! Let us see how Al Khwarizmi leads us through this knot in the Islamic garden of medieval mathematics, and notice on the way how geometry forms the paths in the garden, leading us through the dancing flowers of symbols, signs and formulae.

We start with the equation as Al Khwarizmi worked with it:

x² + bx = c

where c is the area of the whole rectangle.


Next Al Khwarizmi chops the bx term in half, resulting in two rectangles (b/2)x, which he then cunningly rearranges along the edges of the square of side x (x²):
x² + 2(b/2)x = c


If we add a little blue square, top right in the diagram, we can make one big square.


The area of the little blue square is (b/2)². The area of the big square is (x+b/2)². The area of the big square is also equal to (the little blue square + c), which = ((b/2)² + c).

That is to say: (x+b/2)² = ((b/2)² + c)

Therefore (x+b/2) = ±√((b/2)² + c)

Therefore x = -b/2 ±√((b/2)² + c)

= -b/2 ±√(b²/4 + c)

= -b/2 ±√(b²/4 + 4c/4)

= -b/2 ±√(b² + 4c)/2

Therefore x = (-b±√(b² + 4c))/2

This is, as you can see, very close to the quadratic equation formula quoted above, except that we have no 'a' term and we have a plus instead of a minus in the (b² + 4c) part of the formula.

The reason for the plus instead of the minus is that Al Khwarizmi starts with the equation in the form:

x² + bx = c

Whereas we start with:

ax² + bx + c = 0

The interested reader may now obtain the modern formula from Al Khwarizmi's formula by dividing the starting equation (ax² + bx - c = 0) through by a and then substituting the resulting coefficients (the a, b and c terms) into Al Khwarizmi's formula.

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