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.
#1 What values of x would make the following equation true
?
Here are the steps you should follow as you learn this factoring method.
Step 1:
Square
half the
coefficient
of the “x” term and add to both sides:
which yields
Step 2:
Factor
the complete square:
Step 3: Take the
square root
of both sides:
What is your answer?
#2 What values of x would make the following equation true
?
What is your answer?
#3 What values of x would make the following equation true
?
It is easier to do this if the
coefficient
of x
^{2}
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.
#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?
#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?
#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?
#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 1:
Square
half the
coefficient
of the “x” term and add to both sides:
Step 2:
Factor
the complete square:
Step 3: Solve for y:
What is your answer?
#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 x
^{2}
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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