Throughout the years, the history of mathematics has taken its fair share of changes. It has stretched across the world from the Far East, migrating into the Western Hemisphere. One of the most fundamental and key principles of mathematics has been the quadratic formula. Having been used in several different cultures, the formula has been part of the base of mathematics theory. The general equation has been derived from many different sources, most commonly: ax2 + bx + c = 0, with x being the variable and a, b, and c its respective constant terms. Though this is how modern mathematics perceives the equation, different symbols and notations have been used to represent the formula.

Beginning in the “Before Christ” era, the Babylonians were the first to have been recorded demonstrating the equation, circa 400 BC. The form most mathematics students use today is:

To solve a quadratic equation the Babylonians essentially used the standard formula, with the a term being included in the x2 variable. They considered two types of quadratic equations, namely:

x2 + bx = c and x2 - bx = c

Here b and c were positive but not necessarily integers. The form that their solutions took was, respectively:

x = [(b/2)2 + c] - (b/2) and x = [(b/2)2 + c] + (b/2).

Notice that in each case this is the positive root from the two roots of the quadratic and the one that will make sense in solving "real" problems. For example problems which led the Babylonians to equations of this type often concerned the area of a rectangle. For example if the area is given and the amount by which the length exceeds the width is given, then the width satisfies a quadratic equation and then they would apply the first version of the formula above (website one).

The efforts the Babylonians made at using this method were far from futile and, actually, served a very important purpose.

It was an important task for the rulers of Mesopotamia to dig canals and to maintain them, because canals were not only necessary for irrigation but also useful for the transport of goods and armies. The rulers or high government officials must have ordered Babylonian mathematicians to calculate the number of workers and days necessary for the building of a canal, and to calculate the total expenses of wages of the workers.

There are several Old Babylonian mathematical texts in which various quantities concerning the digging of a canal are asked for. They are YBC 4666, 7164, and VAT 7528, all of which are written in Sumerian ..., and YBC 9874 and BM 85196, No. 15, which are written in Akkadian ... . From the mathematical point of view these problems are comparatively simple (Muroi).

The Babylonians used their mathematics not in the way we do today, by teaching, but by building their civilization into what it became. Muroi speaks of how Babylonian mathematics helped create a society and how it helped the Mesopotamian region become as fertile as it was. It is for this reason that they only used positive forms in their answers. Had their use for mathematics been for reasons other than land, monetary, and other non-scientific reasons, they would have had to conclude their method with a negative result.

Moving along the historical timeline, the Greeks were the next prominent mathematics society. Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements. The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. However little is known of Euclid\'s life except that he taught at Alexandria in Egypt. Proclus, the last major Greek philosopher, who lived around 450 AD wrote:

Not much younger than these [pupils of Plato] is Euclid, who put together the "Elements", arranging in order many of Eudoxus\' theorems, perfecting many of Theaetetus\'s, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to

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