# Earliest methods used to solve quadratic

They made tables of many things which allowed them to develop their maths further than previous civilisations, and to calculate things like square roots with as much accuracy as mathematicians in the times of the Renaissance.

## Euclid quadratic formula

In Ren Descartes published La Gomtrie containing the quadratic formula in the form we know today. They made tables of many things which allowed them to develop their maths further than previous civilisations, and to calculate things like square roots with as much accuracy as mathematicians in the times of the Renaissance. For example if the area is given and the amount by which the length exceeds the breadth is given, then the breadth satisfies a quadratic equation and then they would apply the first version of the formula above. Scipione dal Ferro held the Chair of Arithmetic and Geometry at the University of Bologna and certainly must have met Pacioli who lectured at Bologna in They could extract square and cube roots, work with Pythagorean triples years before Pythagoras, had a knowledge of pi and possibly e the exponential function , could solve some quadratics and even polynomials of degree 8, solved linear equations and could also deal with circular measurement. Dal Ferro is credited with solving cubic equations algebraically but the picture is somewhat more complicated. This would not be found for well over three thousand years.

Babylonian mathematical texts are plentiful and well edited. Cardan noticed something strange when he applied his formula to certain cubics.

Dal Ferro is credited with solving cubic equations algebraically but the picture is somewhat more complicated. Babylonian mathematics went far beyond arithmetical calculations. They didn't always use this method though; sometimes it was just as simple to multiply and add, e. Harriot also had a nice method for solving cubics.

In respect of content there is scarcely any difference between the two groups of texts. In this article we now examine some algebra which the Babylonians developed, particularly problems which led to equations and their solution.

### Babylonian quadratic equation

It was the first true algebraic proof of the formula, all previous proofs being geometrical in nature. In it he made many contributions to the understanding of cubic equations. Thus division was a lot easier than the rather messy duplation method of the Egyptians and made arithmetical calculations much easier to carry out. The earliest Egyptian texts, composed about BC, reveal a decimal numeration system with. Accuracy in these kinds of computations was quite easy with the fractional notation they had, and approximations to other Early Equations The earliest record of the quadratic that we know of dates back to the Babylonians, solved on a tablet. Scipione dal Ferro held the Chair of Arithmetic and Geometry at the University of Bologna and certainly must have met Pacioli who lectured at Bologna in From this subtract 3; 30 to give the answer 5 for the breadth of the triangle. An example of a problem of this type is the following. The irreducible case of the cubic, namely the case where Cardan 's formula leads to the square root of negative numbers, was studied in detail by Rafael Bombelli in in his work Algebra.

Related Interests. However, 'logarithm tables' were not used for general calculation but were only used to solve specific problems. Take its square root from a table of squares to get 8; This is how most of their texts have come down to us: as symbols written on wet clay tablets which were then baked in the hot sun so the clay set and the symbols were permanent.

They made tables of many things which allowed them to develop their maths further than previous civilisations, and to calculate things like square roots with as much accuracy as mathematicians in the times of the Renaissance.

In Ars Magna Cardan gives a calculation with 'complex numbers' to solve a similar problem but he really did not understand his own calculation which he says is as subtle as it is useless.

## Famous quadratic equations

The Babylonians had a sexagesimal system and used the concept of place value to write numbers larger than Babylonian cuneiform tablets contain problems reducible to solving quadratic equations. However, 'logarithm tables' were not used for general calculation but were only used to solve specific problems. Thus division was a lot easier than the rather messy duplation method of the Egyptians and made arithmetical calculations much easier to carry out. Apolinar Submitted to: Mr. The problem posed by the scribe is to find the quantity of barley. If linear problems are found in their texts then the answers are simply given without any working; these problems were obviously thought too elementary for much attention. Square and Cube Roots It is quite amazing to find, on an ancient stone tablet, a highly accurate approximation to the square root of 2. The Indian mathematician Brahmagupta AD explicitly described the quadratic formula in his treatise Brhmasphuasiddhnta published in AD, but written in words instead of symbols. There mathematics was dominated by arithmetic, with an emphasis on measurement and calculation in geometry and with no trace of later mathematical concepts such as axioms or proofs. In Ars Magna Cardan gives a calculation with 'complex numbers' to solve a similar problem but he really did not understand his own calculation which he says is as subtle as it is useless. In this article we now examine some algebra which the Babylonians developed, particularly problems which led to equations and their solution. In the notation introduced earlier, we can use a semi-colon to indicate a decimal point. Ferrari managed to solve the quartic with perhaps the most elegant of all the methods that were found to solve this type of problem.

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