Quadratic Equations: Completing the Square

First recall the algebraic identities


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


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


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


Example: Use Complete the Square Method to solve


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


which equivalent to


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


Easy algebraic calculations give


Taking the square-roots lead to


which give the solutions to the equation


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.

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Author: Mohamed Amine Khamsi

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