Completing the Square
There are two ways to complete the square:
The logical way...
And the much more effective mathematical way...
When it comes to deciding what the best method is when solving a problem involving a quadratic function, there are two things that need to be determined initially:
A) Is it in true quadratic form?
and B) Are "a", "b", and "c" all real numbers with "a" not equivalent to zero?
Once these two pieces have been determined, the next step is to analyze the function in order to determine if the factors are friendly numbers, in that they are easily factored. If this is not the case, there are two popular options:
1. The Quadratic Formula
2. Completing the square
No matter which method is used, both result in identical answers in regards to their roots. The advantage to completing the square is that it becomes much easier to set the equation up in what some people refer to as the standard form of a quadratic function (see below). In this equation, point (h, k) represents the vertex of the parabola, x = h represents the vertical axis of the parabola, and "a" determines whether or not the parabola opens upward or downward (positive resulting in upwards and vice versa):
The following steps must be taking in completing the square:
Consider the quadratic function: 
1. Set aside the constant from the right side of the equation an focus solely on the other terms.
2. Determine 
.
.
3. Add and subtract 
within the original parentheses of the quadratic function.
within the original parentheses of the quadratic function.
4. Reorganize the terms and simplify (don't forget to bring back the constant).
represents the original quadratic function now in standard form.
The graphical representation:
- Is a parabola
- Opens upward as "a" = 1
- Has a vertex of (4, -12)
Sidenote: If it is necessary to determine the roots of a quadratic function such as the one seen above, it would be easiest to take its simplest form (original equation) and apply the quadratic formula.
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