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 2*e*=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 2*e*=*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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