There are general steps and guidelines to follow when completing the square. Two full-length examples of how to Complete the Square are shown below.
General Form: f(x) = ax2+bx+c
Step 1: Set the equation equal to zero.
Step 2: Proceed to get rid of the parameter ‘c’ by adding or subtracting the opposite.
Step 3: If there is a co-efficient to the parameter ‘a’, divide both sides of the equation by that number. To clarify, if the number in front of ‘a’ is anything other than 1 (which is equal to itself), then divide both sides of the equation to rid of the number.
Step 4: Now all that should be left are the parameters ‘a’ and ‘b’. Take the co-efficient of ‘b’ and then square it. Half of the co-efficient of ‘b’ will be used later on. For now, we will use the resulted number squared and add it to both sides of the equation.
Step 5: To clarify, the equation should now look like the general form. Now, we will condense this equation into factored form, by combining ½ of ‘b’ and the variable x. This will look identical to (x + half of the co-efficient ‘b’).
Step 6: This step is originally a part of Step 5, however, to break things down, these two were divided. Now, square what’s inside of the parenthesis: (x + number)2.
Step 7: Now, square root both sides of the equation.
Step 8: If the result of ‘f(x)’ should include both the positive and the negative labeling. Next, isolate ‘x’ on the other side of the equation. (By adding or subtracting the opposite.)
Step 9: There should now be two answers for ‘x’, for the positive/negative sign in place of ‘f(x)’ serves as two possible answers. In essence, this last step will create two points along the x-axis for a parabola.
Example Problem 1:
These two answers for x will begin to shape the parabola.
Application
By examining the information above as well as the guide, close attention is needed to identify that in the general form, there is a numeric value for each variable. (If the variable seems not to exist, then the numeric value is zero.)
In the problem above:
a = 1
b = 4
c = -12
When creating parabolas, it is also important to identify whether or not ‘a’ is positive, for that will tell the direction of the parabola. If a > o, the parabola will curve upward. If
a < 0, the parabola will curve downward. Below is the graphed equation from Example Problem One.
(Click For Enlarged View)
Example Problem 2:
Application
By referring to the same process above when Completing the Square and analyzing information, the details of how the parabola is created shall be clear. By the answer requiring some sort of thought for application, this will test your knowledge as to how much you've learned from this skill. Below is Example Problem Two graphed.
(Click For Enlarged View)
