, from an incomplete expression, or 
. The perfect square trinomial can then be written as
a squared quantity 
, which is very useful in a number of situations.
[Back] [VVHS Math - Mr. Davis]
"Completing the square" refers to the process of creating a perfect trinomial square, an expression in the form
, from an incomplete expression, or . The perfect square trinomial can then be written as
a squared quantity , which is very useful in a number of situations.
The first basic idea of completing the square is, therefore,
Any expression in the form 
can be completed to make a perfect square 
.
So, it's very important to memorize the basic model: 
The model shows that the value of a is one-half b or 
and therefore 
completes the square.
1-4 Fill in the blanks in order to make a true equation.
1. 
2. 
3. 
4. 
More exercises would be helpful, but the basic pattern can be seen here.
Solutions to 1-4
1. 
2. 
3. 
4. 
Typically this method is only used on quadratics in the form 
where b is even. It is completely
general, though, and can be used on any quadratic equation. The basic model above is used to create a squared
quantity, which is the only way to isolate and solve for 
in quadratic equations. The quadratic formula is
derived, in fact, from completing the square in the general case.
Study the following examples:
Example II-1)
Notice that when completing the square, the equation must be balanced by adding to the right hand side the
quantity which completes the square.
Here's another example, one not immediately a quadratic equation. Also fun with fractions!
Example II-2)
Easy. However, this method becomes cumbersome for more involved quadratic equations. If the
term has a
coefficient, it must be divided out before completing the square, as follows. Note also how sometimes a little
simplifying is necessary when solving any equation, to create a familiar situation.
Example II-3) 
On other hand, real life applications can spew out fairly complex quadratic functions, for example,
, which is a little tedious to work with mechanically. So, we complete the square
to solve, here, once and for all, with the general quadratic as our given equation, and the famous quadratic
formula as the result.
Quadratic Formula: 
The basic idea here is this:
Any expression in the form 
(general quadratic form)
can also be written in the form 
.
This idea can be used to convert a general quadratic function 
to vertex form, or
. The method is the same as used to solve a quadratic equation, but now is done in place, or all
on the right-hand side of the function. The difference here is the balancing step, where we now do the opposite
of whatever effect completing the square has on the expression. Some examples:
Example III-1) 
Since completing the square added 16 to the expression, subtracting 16 keeps it balanced.
Another basic example, remember, fractions are your buddies.
Example III-2) 
If the 
term has a coefficient it must first be factored off of the 
and 
terms. Study the following two
examples carefully.
Example III-3) 
Notice especially the effect of the coefficient. Since the expression is increased by 2(9) by completing the
square, subtracting 2(9) keeps the equation balanced.
In this last example the parabola opens downward ( a < 0 ), which adds another twist to the balancing step.
Example III-4) 
Notice in this example the -3 means that when you add 16 to complete the square, you've actually added a
negative 48 to the expression, so you have to add 48 to balance.
And, just as the quadratic formula results from solving the general case quadratic, we can complete the square
in place one last time, and derive a useful method for finding the vertex of any parabola quickly using
as
the x-coordinate, and plugging that value in to find the y-coordinate.
Vertex Formula(s)
That's it for quadratic functions and completing the square. C' the S' forever! The method pops up from time to
time, whenever a squared quantity is useful.