# 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 to generate a perfect square. Indeed we have

It is not hard to generalize this to any quadratic function of the form . 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 . Therefore, let divide the equation by 2, to get

which equivalent to

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

Easy algebraic calculations give

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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