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The stable form of carbon is carbon 12 and the radioactive isotope carbon 14 decays over time into nitrogen 14 and other particles. The Greeks were able to solve the quadratic equation by geometric methods, and Euclid's (ca. Viète was among the first to replace geometric methods of solution with analytic ones, although he apparently did not grasp the idea of a general quadratic equation (Smith 1953, pp. An alternate form of the quadratic equation is given by dividing (◇) through by : Given a quadratic integer polynomial , consider the number of such polynomials that are factorable over the integers for and taken from some set of integers . 210-290) solved the quadratic equation, but giving only one root, even when both roots were positive (Smith 1951, p. A number of Indian mathematicians gave rules equivalent to the quadratic formula. Amazingly, the sequence for has the recurrence equation Biquadratic Equation, Carlyle Circle, Completing the Square, Conic Section, Cubic Equation, Fermat's Factorization Method, Polynomial Discriminant, Quadratic, Quadratic Formula, Quartic Equation, Quintic Equation, Sextic Equation Abramowitz, M. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. Among his many other talents, Major General Stanley in Gilbert and Sullivan's operetta the Pirates of Penzance impresses the pirates with his knowledge of quadratic equations in "The Major General's Song" as follows: "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of the hypotenuse." The roots can be found by completing the square, This equation is known as the quadratic formula. 325-270 BC) Data contains three problems involving quadratics. For example, for , there are four such polynomials, The following table summarizes the counts of such factorable polynomials for simple and small . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. In his work Arithmetica, the Greek mathematician Diophantus (ca. 850) had substantially the modern rule for the positive root of a quadratic. 1025) gave the positive root of the quadratic formula, as stated by Bhāskara (ca. Plots of the fractions of factorable polynomials for (red), (blue), and (green) are also illustrated above.
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