It seems that in general it is easier to use one of the standard methods, then convert that to a continued fraction, but there are some special cases that do have straightforward direct continued fraction solutions. I was curious as to whether there was a direct way to express the solution of a general quadratic equation as a continued fraction. Note that it is also possible to solve cubic equations geometrically, but only if you use a less favoured construction called "neusis" which requires a marked straight edge (or if you use origami). You can solve quadratic equations directly using straight edge and compass constructions. In ancient times, mathematicians were interested in solving such problems geometrically. Pattern recognition of perfect square trinomials.Guessing (perhaps helped by rational roots theorem).The answer "As many as you like" may sound frivolous, but there are many ways to solve them and you may want to make your own for a particular circumstance. No factoring by grouping and no solving binomials!!! Proceeding: find 2 real roots of f'(x), then, divide them by a = 8.įind 2 real roots knowing the sum(-b = 22), and the product (ac= -104).
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Use the coefficients of a quadratic equation to help decide which method is most appropriate for solving it. Transformed equation: #f'(x) = x^2 - 22x - 104 = 0#. Solve equations that are quadratic in form. This quadratic equation could be solved by factoring, but well use the method of completing the square. The method is called solving quadratic equations by completing the square. It helps avoid the lengthy factoring by grouping and the solving of the 2 binomials.Įxample. The method we shall study is based on perfect square trinomials and extraction of roots.
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When f(x) can be factored, the new Transforming Method (Google) may be the best method to perform. This formula is easier to remember and to compute as compared to the classic formula. However, there is an improved quadratic formula in graphic form (Google) that is interesting to know. the quadratic formula is the obvious choice. When the quadratic equation f(x) = 0 can't be factored.
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The popular factoring AC method, and the new Transforming Method (Socratic, Google Search) Graphing, completing the square, factoring FOIL, quadratic formula, So far, there are 6 methods to solve quadratic functions.