Solving by Completing the Square (Edexcel IGCSE Maths B): Revision Note

Solving by completing the square

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 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 + 5 = ±4

    • x  = -5 ±4

    • x  = -1 or x  = -9

• It also works with numbers that lead to surds

  • The answers found will be in exact (surd) form

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

• If the equation is ax² + bx + c = 0 with a number (other than 1) in front of x²

  • you can divide both sides by a first (before completing the square)

    • For example 3x² + 12x + 9 = 0

    • Divide both sides by 3

      • x² + 4x + 3 = 0

    • Complete the square on this easier equation

• This trick only works when completing the square to solve a quadratic equation

  • i.e. it has an "=0" on the right-hand side

• Don't do this when using completing the square to rewrite a quadratic expression in a new form

  • i.e. when there is no "=0"

  • For that, you must factorise out the a (but not divide by it)

    • $a{x}^2+bx+c=a\left[x^2+\frac{b}{a}x\right]+c$ and so on

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 when completing the square

    • This makes it 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  = -5 ± 4 (as above) 

    • by the quadratic formula,  $x=\frac{-10\pm \sqrt{64}}{2}=-5\pm \frac{8}{2}$ = -5 ± 4 (the same structure)