The way to find a solution x is to locate point X on BC for which ∠AXD = 90°. It follows that DF = a x² + b x + c and, in case, F = D, x is naturally a solution to the quadratic equation a x² + b x + c = 0. Introduce x = tan ω and find F on CD to make ∠AEF right. For example, with c < 0, the segments AB and CD should be pointing in different directions:Ĭhoose point E on BC and let ω be the ∠EAB. I came across a generalization of Descartes' construction in an old translation (1910) of the classic The Theory of Geometric Constructions by August Adler (1863-1923).įollowing Adler, in order to solve the quadratic equation, form a broken right-angled line ABCD, with AB = a, BC = b, and CD = c taking account of the signs of the coefficients. (By being "constructible" we mean a length (of a straight line) segment that could be constructed by straightedge and compass, given a unit length.) In fact, Descartes used this example to introduce his ideas of analytic geometry. This is exactly the kind of operations that produce constructible numbers from constructible numbers. In addition to the four arithmetic operations, the formula includes a square root. (On a more extended discussion of solving and graphing the quadratic equation see the article Graph and Roots of Quadratic Polynomial.) The roots can be found from the quadratic formula: With the leading coefficient a ≠ 0, has two roots that may be real - equal or different - or complex. On Wolfram|Alpha Quadratic Equation Cite this as:įrom MathWorld-A Wolfram Web Resource.Geometric Construction of Roots of Quadratic Equation "The Quadratic Function and Its Reciprocal." Ch. 16 in AnĪtlas of Functions. Cambridge, England:Ĭambridge University Press, pp. 178-180, 1992. Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. "Quadratic and Cubic Equations." §5.6 in Numerical Oxford,Įngland: Oxford University Press, pp. 91-92, 1996. Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. "Quadratic Equations."Īnd Polynomial Inequalities. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. 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. 449-450).Īn alternate form of the quadratic equation is given by dividing (◇) through by : The Persian mathematiciansĪl-Khwārizmī (ca. 1025) gave the positive root of the quadratic formula, as statedīy Bhāskara (ca. 850) had substantially the modern rule for the positive root of a quadratic. Of the quadratic equations with both solutions (Smith 1951, p. 159 Smithġ953, p. 444), while Brahmagupta (ca. (475 or 476-550) gave a rule for the sum of a geometric series that shows knowledge The method of solution (Smith 1953, p. 444). Solutions of the equation, but even should this be the case, there is no record of It is possible that certain altar constructions dating from ca. 210-290) solved the quadratic equation, but giving only one root, even whenīoth roots were positive (Smith 1951, p. 134).Ī number of Indian mathematicians gave rules equivalent to the quadratic formula. In his work Arithmetica, the Greek mathematician Diophantus The Greeks were able to solve the quadratic equation by geometric methods, and Euclid's (ca.
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