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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク quadratic formula ： ウィキペディア英語版 quadratic formula In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way. The general quadratic equation is :$ax^2+bx+c=0.$ Here represents an unknown, while , , and are constants with not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation, by inserting the former into the latter. Each of the solutions given by the quadratic formula is called a root of the quadratic equation. Geometrically, these roots represent the values at which ''any'' parabola, explicitly given as $y\; =\; ax^2+bx+c,$ crosses the axis. As well as being a formula that will yield the zeros of any parabola, the quadratic equation will give the axis of symmetry of the parabola, and it can be used to immediately determine how many zeros to expect the parabola to have. ==Derivation of the formula== Once a student understands how to complete the square, they can then derive the quadratic formula.〔, (Chapter 13 §4.4, p. 291 )〕〔Li, Xuhui. ''An Investigation of Secondary School Algebra Teachers' Mathematical Knowledge for Teaching Algebraic Equation Solving'', p. 56 (ProQuest, 2007): "The quadratic formula is the most general method for solving quadratic equations and is derived from another general method: completing the square."〕 For that reason, the derivation is sometimes left as an exercise for the student, who can thereby experience rediscovery of this important formula.〔Rockswold, Gary. ''College algebra and trigonometry and precalculus'', p. 178 (Addison Wesley, 2002).〕〔Beckenbach, Edwin et al. ''Modern college algebra and trigonometry'', p. 81 (Wadsworth Pub. Co., 1986).〕 The explicit derivation is as follows. Divide the quadratic equation by , which is allowed because is non-zero: :$x^2\; +\; \backslash frac\; x\; +\; \backslash frac=0.$ Subtract from both sides of the equation, yielding: :$x^2\; +\; \backslash frac\; x=\; -\backslash frac.$ The quadratic equation is now in a form to which the method of completing the square can be applied. Thus, add a constant to both sides of the equation such that the left hand side becomes a complete square: :$x^2+\backslash fracx+\backslash left(\; \backslash frac\; \backslash right)^2\; =-\backslash frac+\backslash left(\; \backslash frac\; \backslash right)^2,$ which produces: :$\backslash left(x+\backslash frac\backslash right)^2=-\backslash frac+\backslash frac$. Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain this: :$\backslash left(x+\backslash frac\backslash right)^2=\backslash frac.$ The square has thus been completed. Taking the square root of both sides yields the following equation: :$x+\backslash frac=\backslash pm\backslash frac.$ Isolating gives the quadratic formula: :$x=\backslash frac.$ The plus-minus symbol "±" indicates that both :$x=\backslash frac\backslash quad\backslash text\backslash quad\; x=\backslash frac$ are solutions of the quadratic equation. There are many alternatives of this derivation with minor differences, mostly concerning the manipulation of $a$. Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as or , where has a magnitude one half of the more common one. These result in slightly different forms for the solution, but are otherwise equivalent. A lesser known quadratic formula, as used in Muller's method, and which can be found from Vieta's formulas, provides the same roots via the equation: :$x=\backslash frac\{b\backslash pm\backslash sqrt\{b^2-4ac\}\}.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「quadratic formula」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.