Solving by Completing the Square (Edexcel IGCSE Maths)

Spec point — 2.7B

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 + 10*x *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

Exam Hint When making x the subject to find the solutions, don't expand the squared bracket back out again!

• Remember to use ±√ to get two solutions

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)

    • $a x^{2} + b x + c = a [ x^{2} + \frac{b}{a} x ] + c$ 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)

Worked Example Solve $2x^{2}-8x-24=0$ by completing the square.

Answer:

Divide both sides by 2 to make the quadratic start with x²* *

$x^{2}-4x-12=0$

Halve the middle number, -4, to get -2 Replace the first two terms, x² - 4**x, with (x - 2)²- (-2)²

$( x - 2 )^{2} - ( - 2 )^{2} - 12 = 0$

Simplify the numbers

$\begin{array}{rcl} ( x - 2 )^{2} - 4 - 12 & = 0 \\ ( x - 2 )^{2} - 16 & = 0 \end{array}$

Add 16 to both sides

$( x - 2 )^{2} = 16$

Take the square root of both sides Include the ± sign to get two solutions

$x-2=\pm\sqrt{16}=\pm4$

Add 2 to both sides

$x=2\pm4$

Work out each solution separately

Final answer: x = 6 or x = -2