Quadratic Equations: Completing the Square

First recall the algebraic identities

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We shall use these identities to carry out the process called Completing the Square. For example, consider the quadratic function

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What can be added to yield a perfect square? Using the previous identities, we see that if we put 2e=8, that is e=4, it is enough to add tex2html_wrap_inline70 to generate a perfect square. Indeed we have

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It is not hard to generalize this to any quadratic function of the form tex2html_wrap_inline74 . In this case, we have 2e=b which yields e=b/2. Hence

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Example: Use Complete the Square Method to solve

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Solution.First note that the previous ideas were developed for quadratic functions with no coefficient in front of tex2html_wrap_inline84 . Therefore, let divide the equation by 2, to get

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which equivalent to

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In order to generate a perfect square we add tex2html_wrap_inline90 to both sides of the equation

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Easy algebraic calculations give

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Taking the square-roots lead to

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which give the solutions to the equation

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We have developed a step-by-step procedure for solving a quadratic equation; or, in other words, an algorithm for solving a quadratic equation. This algorithm can be stated as a formula called Quadratic Formula.

[Algebra] [Complex Variables]
[Geometry] [Trigonometry ]
[Calculus] [Differential Equations] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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