Completing the Square

There are  times it is necessary to convert a polynomial into a perfect square.

Step 1 - Understanding the Problem
Let's start by reviewing how to square a binomial. (hint use FOIL: First-Outer-Inner-Last)

(a + b)2 = (a + b)(a + b) = a2 + ab + ab + b = a2 + 2ab + b2

Notice that the middle term of a2 + 2ab + b2    is equal to 2 times the product of the first term , a , and last term , b , of the original binomial. 2ab
So to complete the square, we want to perform the opposite operation on the expression. Take half the coefficient of the middle term ....and square it
Here is the proof:

 Show this is true!

 

Step 2 - Complete the Square
Example 1
We want to convert the expression x2 + 6x  into a perfect square.
Take half the coefficient of the middle term ....and square it
 x2 + 6x + (6/2)2 = x2 + 6x + 32 = x2 + 6x + 9 = 
(x + 3)(x + 3) = (x + 3)2
 
Example 2
Convert the expression x2 + 10x  into a perfect square.
x2 + 10x + (10/2)2 = x2 + 10x + 52 = x2 + 10x + 25 =
(x + 5)(x + 5) = (x + 5)2
 
Example 3
Convert the expression x2 - 5x  into a perfect square.
x2 - 5x + (5/2)2 = x2 - 5x + (5/2)2 = x2 - 5x + 25/4 =
(x - 5/2)(x - 5/2) = (x - 5/2)2
Notice: 
(a - b)2 = (a - b)(a - b) = a2 - ab - ab + b = a2 - 2ab + b2
 
Example 4  
Convert the expression x2 - 5x  into a perfect square.
x2 - 5x + (5/2)2 = x2 - 5x + (5/2)2 = x2 - 5x + 25/4 =
(x - 5/2)(x - 5/2) = (x - 5/2)2
 
Example 5 - Standard form of a Circle   
Given:  x2y2 + 10x - 3y  + 3 = 0
Write in standard form: (x - h)2 + (x - k)2 = r2 
x2y2 + 10x - 3y  + 3 = 0 Group like terms
x2 + 10xy2 - 3y  = -3 Reorder and subtract 3
x2 + 10x52 + y2 - 3y  + (3/2)2 = -3 + 52 + (3/2)2 Take half the middle term and square it. Note half of 3 is 3/2
(x + 5)2 + (x - 3/2)2  =  -3 + 25 + 9/4 Complete the squares
(x + 5)2 + (x - 3/2)2  = 107/4
Here the center is (-5,3/2) Radius = sqrt(107)/2