Quadratic Equations: Completing the Square
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
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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