Completing the Square (Cambridge O Level Maths)

How can I rewrite the first two terms of a quadratic expression as the difference of two squares?

• Look at the quadratic expression x² + bx + c 
• The first two terms can be written as the difference of two squares using the following rule

is the same as where is half of 

• Check this is true by expanding the right-hand side
  • Is the same as ?
    
    • Yes: (x + 1)(x + 1) - 1² = x² + 2x + 1 - 1 = x² + 2x
• This works for negative values of b too
  •   can be written as  which is 
  • A negative b does not change the sign at the end

How do I complete the square?

• Completing the square is a way to rewrite a quadratic expression in a form containing a squared-bracket
• To complete the square on x² + 10x + 9
  • Use the rule above to replace the first two terms, x² + 10x, with (x + 5)² - 5²
  • add 9:  (x + 5)² - 5² + 9
  • simplify the numbers:  (x + 5)² - 25 + 9
  • answer: (x + 5)² - 16 

How do I complete the square when there is a coefficient in front of the x² term?

• You first need to take out as a factor of the x² and x terms only
  • 
    • Use square-shaped brackets here to avoid confusion with curly brackets later
• Then complete the square on the bit inside the square-brackets:
  • This gives 
    • where p is half of 
• Finally multiply this expression by the a outside the square-brackets and add the c
  • 
  • This looks far more complicated than it is in practice!
    • Usually you are asked to give your final answer in the form   
• For quadratics like , do the above with a = -1

How do I find the turning point by completing the square?

• Completing the square helps us find the turning point on a quadratic graph
  • If  then the turning point is at
    • Notice the negative sign in the x-coordinate
    • This links to transformations of graphs (translating by p to the left and q up)
  • If  then the turning point is still at 
    • It's at a minimum point if a > 0
    • It's at a maximum point if a < 0
• It can also help you create the equation of a quadratic when given the turning point

• It can also be used to prove and/or show results using the fact that any "squared term", i.e. the bracket (x ± p)², will always be greater than or equal to 0
  • You cannot square a number and get a negative value

How do I solve a quadratic equation by completing the square?

• To solve x² + bx + c = 0 
  • replace the first two terms, x² + bx, with (x + p)² - p² where p is half of b
  • this is called completing the square
    • x² + bx + c = 0 becomes
      • (x + p)² - p² + c = 0 where p is half of b
  • rearrange this equation to make x the subject (using ±√)
• For example, solve x² + 10x + 9 = 0 by completing the square
  • x² + 10x becomes (x + 5)² - 5²
  • so x² + 10x + 9 = 0 becomes (x + 5)² - 5² + 9 = 0
  • make x the subject (using ±√)
    
    • (x + 5)² - 25 + 9 = 0
    • (x + 5)² = 16
    • x + 5 = ±√16
    • x  = ±4 - 5
    • x  = -1 or x  = -9

• If the equation is ax² + bx + c = 0 with a number in front of x², then divide both sides by a first, before completing the square 

How does completing the square link to the quadratic formula?

• The quadratic formula actually comes from completing the square to solve ax² + bx + c = 0
  • a, b and c are left as letters, to be as general as possible
• You can see hints of this when you solve quadratics 
  • For example, solving x² + 10x + 9 = 0 
    • by completing the square, (x + 5)² = 16 so x  = ± 4 - 5 (from above) 
    • by the quadratic formula,  = ± 4 - 5 (the same structure)