Sunday, June 28, 2009

The Solution of the Quadratic Equation

We learn quite early in our mathematical education that if we consider the equation (called a quadratic)

then x has the two values

But have you ever wondered how this came about, who discovered it and why it was even considered worthy of attention in the first place? It is an interesting story that goes back at least 4000 years.

In the ancient world, mathematics was primarily a tool that was used to solve everyday problems in building and trade. A problem frequently encountered by retailers and merchants was (and still is today!) of the type

“If 6 rolls of linen cost $20, what do 8 cost?”

The approach to solving this sort of problem usually involves working out the cost of one unit and then just multiplying by the number of units required.

So we let x be the cost of one unit and we can say therefore that

6x = $20

and so

x = $20/6 = $3.33

The cost of 8 units is therefore

$8x = $26.64

This is an example of what are called “linear equations” which take the form

ax – b = 0,

and which have the solution

x = b/a (or 20/6 in our example above)

Linear equations and their solutions were well known to the Babylonians and Egyptians of the period around 2500 BC, and we know this from engraved stone tablets that have been found from the area. (Babylon was the site of one of the most advanced of the ancient civilisations and was located not far from present day Baghdad)

But the need arose to find the solution to another type of equation that was more complicated. This type of equation always emerged in problems involving areas of land, and was therefore very important in dealings concerning real estate.

This sort of problem was of the type

“We want to fence in a rectangular field of 10 square units. One side of the rectangle is 4 units less than the other side. How long are the sides of the field?”

If we say x is the length of the long side, mathematically we can express this statement as

, or


and the problem is to find x.

This type of equation (called a quadratic) is very different to a linear equation, and there is no obvious way of finding a solution.

Precisely when and who made the breakthrough is not known, but sometime in the period around 2000 BC, and probably in Babylon, a very clever technique called “substitution” was employed which finally solved the riddle of the quadratic equation. Here’s how it works.

Recalling again our general quadratic equation


we now introduce another variable y, where


Substituting this back into (2) and rearranging, we find that



From (3) we see that

and substituting this back into (4), followed by further rearrangement, we find that

This neat solution was one of the milestones of mathematics, and had a profound influence on the ancient world.

Incidentally you can use this technique to solve equation (1) above and find that the dimensions of our field are about 5.742 and 1.742 units. Multiply these two together and we get our 10 square units as required.

Dirk Struik remarks in his excellent “A Concise History of Mathematics”, that “although the Egyptians of this period (~2000 BC) were only able to solve simple linear equations, the Babylonians were in full possession of the technique of handling quadratic equations”.

Above: A Babylonian mosaic - c 500BC.

The Babylonians were extraordinarily advanced in art and the sciences, in particular mathematics.
Photo: Wikipedia Commons

We can therefore assume that this method has been known for at least 4000 years, but its beauty and elegance still remains fresh with us today.

We tend to think of the ancient world as a place of ignorance and primitive thought, but I wonder how many modern humans could derive this formula, from first principles, today?

For some more complicated mathematics, involving some old work on calculating pi, go to

Reference: “A Concise History of Mathematics” - D.J.Struik, Dover Publications 1967.

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