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To solve a quadratic equation, the Babylonians essentially used the standard quadratic formula. They considered quadratic equations of the form: + = where b and c were not necessarily integers, but c was always positive. They knew that a solution to this form of equation is: [15]
Plimpton 322 is a Babylonian clay tablet, believed to have been written around 1800 BC, that contains a mathematical table written in cuneiform script.Each row of the table relates to a Pythagorean triple, that is, a triple of integers (,,) that satisfies the Pythagorean theorem, + =, the rule that equates the sum of the squares of the legs of a right triangle to the square of the hypotenuse.
In stage 2, the well-attested Old Babylonian method of completing the square is used to solve what is effectively the system of equations b − a = 0.25, ab = 0.75. [6] Geometrically this is the problem of computing the lengths of the sides of a rectangle whose area A and side-length difference b − a are known, which was a recurring problem ...
YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world". [ 1 ]
An Old Babylonian tablet (Strasbourg 363) seeks the solution of a quadratic equation. [1] c. 1800 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script. [2] 1800 BC: Berlin Papyrus 6619 (19th dynasty) contains a quadratic equation and its solution. [3] [4] 800 BC
Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed. Quadratic equations played an important role in early algebra; and throughout most of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories.
The tablets also include multiplication tables and methods for solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation of √ 2 that is accurate to an equivalent of six decimal places. Babylonian mathematics were written using a sexagesimal (base-60) numeral system.
Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.