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: FirstOuterInnerLast)}  
(a + b)^{2} = (a + b)(a + b) = a^{2} + ab + ab + b^{2 } = a^{2} + 2ab + b^{2} 

Notice that the middle term of a^{2} + 2ab + b^{2 } 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 x^{2} +
6x^{ } into
a perfect square. Take half the coefficient of the middle term ....and square it 

x^{2} +
6x
+ (6/2)^{2}
= x^{2} + 6x + 3^{2}
= x^{2} + 6x + 9 = (x + 3)(x + 3) = (x + 3)^{2} 

^{Example 2}  
Convert the expression x^{2} + 10x^{ } into a perfect square.  
x^{2} +
10x
+ (10/2)^{2}
= x^{2} + 10x + 5^{2}
= x^{2} + 10x + 25 = (x + 5)(x + 5) = (x + 5)^{2} 

^{Example 3}  
Convert the expression x^{2}  5x^{ } into a perfect square.  
x^{2}
 5x
+ (5/2)^{2}
= x^{2}  5x + (5/2)^{2}
= x^{2}  5x + 25/4 = (x  5/2)(x  5/2) = (x  5/2)^{2} 

Notice: (a  b)^{2} = (a  b)(a  b) = a^{2}  ab  ab + b^{2 } = a^{2}  2ab + b^{2} 

^{Example 4 }  
Convert the expression x^{2}  5x^{ } into a perfect square.  
x^{2}
 5x
+ (5/2)^{2}
= x^{2}  5x + (5/2)^{2}
= x^{2}  5x + 25/4 = (x  5/2)(x  5/2) = (x  5/2)^{2} 

^{Example 5  Standard form of a Circle }  
Given: x^{2}
+ y^{2} + 10x  3y + 3 = 0 Write in standard form: (x  h)^{2} + (x  k)^{2} = r^{2} 

x^{2} + y^{2} + 10x  3y + 3 = 0  Group like terms 
x^{2} + 10x + y^{2 } 3y = 3  Reorder and subtract 3 
x^{2} + 10x + 5^{2 } + y^{2 } 3y + (3/2)^{2 } = 3 + 5^{2 } ^{ }+ (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 
Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada