[<< wikibooks] Algebra/Completing the Square
== Derivation ==
The purpose of "completing the square" is to either factor a prime quadratic equation or to more easily graph a parabola. The procedure to follow is as follows for a quadratic equation 
  
    
      
        y
        =
        a
        
          x
          
            2
          
        
        +
        b
        x
        +
        c
      
    
    {\displaystyle y=ax^{2}+bx+c}
  :
1. Divide everything by a, so that the number in front of 
  
    
      
        
          x
          
            2
          
        
      
    
    {\displaystyle x^{2}}
   is a perfect square (1):

  
    
      
        
          
            y
            a
          
        
        =
        
          x
          
            2
          
        
        +
        
          
            b
            a
          
        
        x
        +
        
          
            c
            a
          
        
      
    
    {\displaystyle {\frac {y}{a}}=x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}}
  2. Now we want to focus on the term in front of the x. Add the quantity 
  
    
      
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
      
    
    {\displaystyle \left({\frac {b}{2a}}\right)^{2}}
   to both sides:

  
    
      
        
          
            y
            a
          
        
        +
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
        =
        
          x
          
            2
          
        
        +
        
          
            b
            a
          
        
        x
        +
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
        +
        
          
            c
            a
          
        
      
    
    {\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}
  3. Now notice that on the right, the first three terms factor into a perfect square:

  
    
      
        
          x
          
            2
          
        
        +
        
          
            b
            a
          
        
        x
        +
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
        =
        
          
            (
            
              x
              +
              
                
                  b
                  
                    2
                    a
                  
                
              
            
            )
          
          
            2
          
        
      
    
    {\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}}
  Multiply this back out to convince yourself that this works.
4. Therefore the completed square form of the quadratic is:

  
    
      
        
          
            y
            a
          
        
        +
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
        =
        
          
            (
            
              x
              +
              
                
                  b
                  
                    2
                    a
                  
                
              
            
            )
          
          
            2
          
        
        +
        
          
            c
            a
          
        
      
    
    {\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}
   or, multiplying through by a,


== Explanation of Derivation ==
 
1. Divide everything by a, so that the number in front of 
  
    
      
        
          x
          
            2
          
        
      
    
    {\displaystyle x^{2}}
   is a perfect square (1):

  
    
      
        
          x
          
            2
          
        
        +
        
          
            b
            a
          
        
        x
        +
        
          
            c
            a
          
        
        =
        
          a
        
      
    
    {\displaystyle x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}={a}}
  Think of this as expressing your final result in terms of 1 square x.  If your initial equation is 

2. Now we want to focus on the term in front of the x. Add the quantity 
  
    
      
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
      
    
    {\displaystyle \left({\frac {b}{2a}}\right)^{2}}
   to both sides:

  
    
      
        
          
            y
            a
          
        
        +
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
        =
        
          x
          
            2
          
        
        +
        
          
            b
            a
          
        
        x
        +
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
        +
        
          
            c
            a
          
        
      
    
    {\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}
  
3. Now notice that on the right, the first three terms factor into a perfect square:

  
    
      
        
          x
          
            2
          
        
        +
        
          
            b
            a
          
        
        x
        +
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
        =
        
          
            (
            
              x
              +
              
                
                  b
                  
                    2
                    a
                  
                
              
            
            )
          
          
            2
          
        
      
    
    {\displaystyle x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}}
  Multiply this back out to convince yourself that this works.
4. Therefore the completed square form of the quadratic is:

  
    
      
        
          
            y
            a
          
        
        +
        
          
            (
            
              
                b
                
                  2
                  a
                
              
            
            )
          
          
            2
          
        
        =
        
          
            (
            
              x
              +
              
                
                  b
                  
                    2
                    a
                  
                
              
            
            )
          
          
            2
          
        
        +
        
          
            c
            a
          
        
      
    
    {\displaystyle {\frac {y}{a}}+\left({\frac {b}{2a}}\right)^{2}=\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}}
   or, multiplying through by a,


== Example ==
The best way to learn to complete a square is through an example. Suppose you want to solve the following equation for x.