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Completing The Square
A “complete” square is a quadratic expression such as which can be factored as the square of a term. In this case we would have

An expression such as  is not a complete square because it cannot be factored as the square of a term as we had with the previous expression .

Knowing how to complete a square can be of assistance in solving equations and writing certain equations in standard form. Examples of each are shown below.

Example Group #1
Here are three examples of using the technique of completing the square to solve equations for x.
Example
#1 What values of x would make the following equation true ?

Here are the steps you should follow as you learn this factoring method.
which yields
  • Step 2: Factor the complete square:
What is your answer?
 
Example
#2 What values of x would make the following equation true ?
What is your answer?
 
Example
#3 What values of x would make the following equation true ?

It is easier to do this if the coefficient of x2 is 1. So you should first divide both sides by 2: . Now follow the four steps outlined above in Example #1 to solve for x.

What is your answer?
 

Example Group #2
Here are four examples of using the technique of completing the square to determine the standard form of each of the classic conic sections: circles, ellipses, parabolas, and hyperbolas.
Example
#1 Use the technique of completing the square to determine the center and radius for this circle:
  • Step 1: Square half the coefficient of the “x” and “y” terms and add to both sides:
  • Step 2: Factor the complete squares:
What is your answer?
 
Example
#2 For the ellipse determine the co-ordinates of its center and then find the lengths of its semi-major axis, a; its semi-minor axis, b; and the distance from each focus to the center, c.
  • Step 1: Factor the 9 and the 4 from the terms in x and y:

  • Step 2: Square half the coefficient of the “x” and “y” terms within the parentheses.  To add to the right side, note that you must multiply by 9 and 4 first.
  • Step 3: Factor the complete squares:
  • Step 4: Divide both sides by 36:
What is your answer?
 
Example
#3 For the hyperbola determine the co-ordinates of its center and then find the lengths of its semi-major axis, a; semi-minor axis, b; and the distance from its center to each focus, c.
What is your answer?
 
Example
#4 For the parabola determine the co-ordinates of its vertex and whether it opens up or down.

As given, this equation is not a complete square.  By completing the square, this equation can be rewritten in “vertex” form as follows:
  • Step 2: Factor the complete square:
  • Step 3: Solve for y:
What is your answer?
 
Example
#5 For the parabola determine the co-ordinates of its vertex and whether it opens up or down.
  • It is easier to do this if the coefficient of x2 is 1. So you should first factor -3 from the terms in x:
  • Step 2: Square half the coefficient of the “x” term within the parentheses. To add to the left side, note that you must multiply by -3 first.
  • Step 3: Factor the complete square:
  • Step 4: Solve for y:
What is your answer?
 



M Ransom

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