Completing the Square (CIE IGCSE Additional Maths)

What is completing the square?

• Completing the square is another way of writing a quadratic function
• It means rewriting in the form 
  • The key point is that now only occurs once in the equation
• It can be used to solve quadratic equations, sketch their graphs and to find the coordinates of the turning point

How do I complete the square?

The method used will depend on the value of the coefficient of the term in 

• When
  • is half of
  • is 

• When a ≠ 1 
  • First take a factor of out of the  and  terms
  • Then continue as above

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