Completing the Square (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Completing the square

What is completing the square?

• Completing the square is another way of writing a quadratic function

• It means rewriting $y=a{x}^2+bx+c$ in the form $y=a{(x+p)}^2+q$

  • The key point is that $x$ 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 ${\mathit{x}}^{\mathbf{2}}$ term in $y=a{x}^2+bx+c$

• When $\mathit{a}\mathbf{=}\mathbf{1}$

  • $p$ is half of $b$

  • $q$ is $c-p^2$

• When a ≠ 1 

  • First take a factor of $a$ out of the ${\mathit{x}}^{\mathbf{2}}$ and $\mathit{x}$ terms

  • Then continue as above

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 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