Earliest Methods used to solve Quadratic Equation

3413 Words Dec 12th, 2014 14 Pages
Earliest Methods used to solve Quadratic Equation
1.
Babylonian mathematics (also known as Assyro-Babylonian mathematics) was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited.[7] In respect of time they fall in two distinct groups: one from the Old Babylonian period (1830-1531 BC), the other mainly Seleucid from the last three or four centuries BC. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia.[7]
In contrast to the scarcity of sources in Egyptian
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The second method uses an algorithm which was later ascribed to the Greeks.
Let a = a1 be an initial approximation. If a1 < sqrt(2) then 2/a1 > sqrt(2). So as a better approximation take a2 = (a1 + 2/a1)/2. Repeat the process until you have an answer as accurate as you want.
Quadratic Equations and the n3 + n2 table
One important table for Babylonian algebra was that of the values of n3 + n2 for integer values of n from 1 to 30. These tables could be used to solve cubic equations of the form ax3 + bx2 = c although note that the Babylonians would not have had this algebraic notation.
Multiplying through by a2/b3 gives:
(ax/b)3 + (ax/b)2 = ca2/b3
Putting y = ax/b gives us the equation y3 + y2 = ca2/b3 which can be solved by looking up in the table to find the value of y and then substituting back.
It is amazing that without the use of modern notation for these equations the Babylonians could recognise equations of a certain type and the methods for solving them.
It is hardly surprising then to find that the Babylonians were also proficient at solving quadratic equations. If linear problems are found in their texts then the answers are simply given without any working; these problems were obviously thought too
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