Completing the square | Year 11

01 / LEARN

The idea, step by step

Create a perfect square

To complete $x^{2} + b x$ , halve b and square the result. Add and subtract that square so the expression stays unchanged.

In ${(x - h)}^{2} + k$ , the square is nonnegative, so k is the minimum. A negative leading coefficient changes this to a maximum.

x^{2} + b x + c = {(x + \frac{b}{2})}^{2} + c - \frac{b^{2}}{4}

Keep the outside coefficient

When the leading coefficient is not 1, factor it out of the quadratic and linear terms first. Complete the square inside, then distribute the coefficient carefully.

When solving, isolate the square. A positive right side gives two branches, zero gives one and a negative right side gives no real solutions.

A common trap

Replacing $x^{2} + 6 x$ with ${(x + 3)}^{2}$ alone.

Instead: The square adds 9: subtract 9 as well.

02 / WORKED EXAMPLES

See the thinking

Example 1

Reveal a minimum

Complete the square in $x^{2} - 8 x + 11$ .

Half -8 is -4; its square is 16.
Write $x^{2} - 8 x + 16 - 16 + 11$ .
Combine constants.

{(x - 4)}^{2} - 5

Example 2

Leading coefficient two

Rewrite $2 x^{2} + 12 x + 5$ and find its minimum.

Factor $2 (x^{2} + 6 x) + 5$ .
Inside, $x^{2} + 6 x = {(x + 3)}^{2} - 9$ .
Hence $2 {(x + 3)}^{2} - 18 + 5$ .

2 {(x + 3)}^{2} - 13

; minimum -13 at x=-3.

Check your understanding

Complete the square: $x^{2} + 10 x + 7$ .

Show answer

${(x + 5)}^{2} - 18$

Add/subtract 25, then combine 7-25.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write $x^{2} + 4 x + 1$ in completed-square form.
2 marks
Write $x^{2} - 6 x + 2$ in completed-square form.
2 marks
Rewrite $3 x^{2} - 12 x + 8$ in square form.
3 marks
Solve ${(x - 2)}^{2} = 7$ exactly.
2 marks
Find the maximum of $- 2 {(x + 1)}^{2} + 9$ .
2 marks
Solve $x^{2} - 4 x - 1 = 0$ by completing the square.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write $x^{2} + 4 x + 1$ in completed-square form.
2 marks
Add/subtract 4.
Group the perfect square.
${(x + 2)}^{2} - 3$
Write $x^{2} - 6 x + 2$ in completed-square form.
2 marks
Add/subtract 9.
Combine constants.
${(x - 3)}^{2} - 7$
Rewrite $3 x^{2} - 12 x + 8$ in square form.
3 marks
Use $3 (x^{2} - 4 x) + 8$ .
Replace the inside with ${(x - 2)}^{2} - 4$ .
$3 {(x - 2)}^{2} - 4$
Solve ${(x - 2)}^{2} = 7$ exactly.
2 marks
Take both roots.
Add 2.
$x = 2 \pm \sqrt{7}$
Find the maximum of $- 2 {(x + 1)}^{2} + 9$ .
2 marks
The square is at least zero.
Its negative multiple is at most zero.
9 at x=-1.
Solve $x^{2} - 4 x - 1 = 0$ by completing the square.
3 marks
Rewrite ${(x - 2)}^{2} - 5 = 0$ .
Isolate the square and take both roots.
$x = 2 \pm \sqrt{5}$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topics 2.1–2.2: quadratic relationships, cubic and quartic polynomials. Original Ascent teaching sequence.

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