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Quadratics – Completing the square

📘 Notes


 

How to Complete the Square


 


   A quadratic expression has the general form:  
   $$ax^2 + bx + c, \quad a \neq 0$$
 

 


   Completing the square means rewriting the quadratic in the form  
   $$(x + p)^2 + q$$  
   This form is useful for solving equations and for finding the maximum/minimum values of a quadratic function.
 

 
 

Case 1: When the coefficient of \(x^2\) is 1


 

       
  • Step 1: Identify coefficients \(a\), \(b\), and \(c\).

    •    
  • Step 2: Take half the coefficient of \(x\) and square it.

    •    
  • Step 3: Add and subtract this square inside the expression.

    •    
  • Step 4: Rewrite as a perfect square plus/minus a constant.

    •  

 


   

Example 1


   

Question: Rewrite \(x^2 + 5x + 3\) in completed square form.

   

Step 1: \(a = 1,\; b = 5,\; c = 3\).


   

Step 2: Half of \(b = 5\) is \(\tfrac{5}{2}\). Squaring gives \(\left(\tfrac{5}{2}\right)^2 = \tfrac{25}{4}\).


   

Step 3: Add and subtract \(\tfrac{25}{4}\):  
     $$x^2 + 5x + 3 = \left(x^2 + 5x + \tfrac{25}{4}\right) - \tfrac{25}{4} + 3$$  
     $$= \left(x + \tfrac{5}{2}\right)^2 - \tfrac{13}{4}$$
   


   

Final Answer:  
     $$x^2 + 5x + 3 = \left(x + \tfrac{5}{2}\right)^2 - \tfrac{13}{4}$$
   


 

 
 

Case 2: When the coefficient of \(x^2\) is not 1


 


   Factor out the coefficient of \(x^2\) first, then complete the square inside the brackets.
 

 


       
  • Step 1: Factor out the coefficient of \(x^2\).

    •    
  • Step 2: Take half the coefficient of \(x\) inside the bracket and square it.

    •    
  • Step 3: Adjust by subtracting the extra constant, then simplify.

    •    
  • Step 4: Expand back if required.

    •  

 


   

Example 2


   

Question: Rewrite \(3x^2 - 9x + 50\) in completed square form.

   

Step 1: Factor out 3:  
     $$3x^2 - 9x + 50 = 3(x^2 - 3x) + 50$$
   

   

Step 2: Half of \(-3\) is \(-\tfrac{3}{2}\). Squaring gives \(\tfrac{9}{4}\).  
     Add and subtract inside:  
     $$3\left(x^2 - 3x + \tfrac{9}{4}\right) - 3\left(\tfrac{9}{4}\right) + 50$$  
     $$= 3\left(x - \tfrac{3}{2}\right)^2 + \tfrac{173}{4}$$
   

   

Final Answer:  
     $$3x^2 - 9x + 50 = 3\left(x - \tfrac{3}{2}\right)^2 + \tfrac{173}{4}$$
   


 

 

Summary


 


   To complete the square for \(ax^2 + bx + c\):  
   
• Factor out \(a\) if \(a \neq 1\).  
   
• Add and subtract \(\left(\tfrac{b}{2a}\right)^2\).  
   
• Rewrite as \((x + \tfrac{b}{2a})^2 + \text{constant}\).