Completing the Square (Cambridge O Level Additional Maths)

Revision Note

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Author

Mark

Last updated

12 September 2023

Completing the Square

What is completing the square?

Completing the square is another way of writing a quadratic function
It means rewriting $y = a x^{2} + b x + c$ 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 $x^{2}$ term in $y = a x^{2} + b x + c$

When $a = 1$
- $p$ is half of $b$
- $q$ is $c - p^{2}$

Example of completing the square

When a ≠ 1
- First take a factor of $a$ out of the $x^{2}$ and $x$ terms
- Then continue as above

Harder example of completing the square

Examiner Tip

Sometimes a question will explicitly use the phrase complete the square
Sometimes a question will use the form $a {(x + p)}^{2} + q$ without using the phrase completing the square

Worked example

Write $3 x^{2} - 12 x + k$ in the form $a {(x + p)}^{2} + k + q$ , where $a, p$ and $q$ are constants to be found, and $k$ is an unknown constant.

The form required is 'completing the square' (do not be put off by the $k$ , it is just a constant!).

STEP 1 - Take a factor of 3 out of the $x^{2}$ and $x$ terms - leaving $k$ avoids awkward fractions.

$3 x^{2} - 12 x + k = 3 (x^{2} - 4 x) + k$

STEPS 2 and 3 - Complete the square on the $(x^{2} - 4 x)$ part only. " $p = - \frac{4}{2} = - 2$ " and " $q = 0 - {(- 2)}^{2} = 4$ ".

$3 [{(x - 2)}^{2} - 4] + k$

STEP 4 - Expand and simplify.

$3 {(x - 2)}^{2} - 12 + k$

$3 {(x - 2)}^{2} + k - 12$

i.e. $a = 3, p = - 2, q = - 12$

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

Examiner Tip

When making x the subject to find the solutions at the end, don't expand the squared brackets back out again!
- Remember to use ±√ to get two solutions

Worked example

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

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

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

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

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

Square root both sides
Include the ± sign to get two solutions

$x - 2 = \pm \sqrt{16} = \pm 4$

Add 2 to both sides

$x = \pm 4 + 2$

Work out each solution separately

x = 6 or x = -2

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