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Edexcel IGCSE Maths (4MA1) · Unit 3: Equations & Sequences

Edexcel IGCSE 4MA1 Completing the Square: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

Completing the square writes $x^{2} + b x + c$ as ${(x + \frac{b}{2})}^{2} - {(\frac{b}{2})}^{2} + c$ : halve the coefficient of $x$ , write that bracket squared, then subtract the square of the half. The result $(x + p)^{2} + q$ gives the turning point $(- p, q)$ of the graph and lets you solve the equation exactly.

Unit 3: Equations & Sequences
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Notes
Formulas
Worked examples
Mistakes
FAQ

Completing the Square notes

When the coefficient of x² is 1

For $x^{2} + 8 x + 3$ , half of 8 is 4, so $x^{2} + 8 x + 3 = (x + 4)^{2} - 16 + 3 = (x + 4)^{2} - 13$ . With an odd coefficient, keep fractions: $x^{2} - 3 x + 1 = {(x - \frac{3}{2})}^{2} - \frac{5}{4}$ .

When the coefficient of x² is not 1

Take the coefficient out of the $x$ terms only: $2 x^{2} + 12 x + 5 = 2 (x^{2} + 6 x) + 5 = 2 [(x + 3)^{2} - 9] + 5 = 2 (x + 3)^{2} - 13$ .

Turning points

The graph of $y = (x + p)^{2} + q$ has its minimum point at $(- p, q)$ , because a square is never negative. So $y = x^{2} + 8 x + 3 = (x + 4)^{2} - 13$ has its minimum at $(- 4, - 13)$ , and the least value of $x^{2} + 8 x + 3$ is $- 13$ . The graph shows worked example 1: $y = x^{2} + 14 x + 20 = (x + 7)^{2} - 29$ turns at $(- 7, - 29)$ .

Graph of y = x² + 14x + 20 with its minimum point (−7, −29) marked, read from the completed square (x + 7)² − 29 — Worked example 1 drawn with our Grapher: the completed square (x + 7)² − 29 gives the minimum point.

Solving by completing the square

$x^{2} + 8 x + 3 = 0$ becomes $(x + 4)^{2} = 13$ , so $x + 4 = \pm \sqrt{13}$ and $x = - 4 \pm \sqrt{13}$ . Questions often want this exact surd form.

Key formulas

Completing the square Learn it: $x^{2} + b x + c = {(x + \frac{b}{2})}^{2} - \frac{b^{2}}{4} + c$
Turning point of $y = (x + p)^{2} + q$ Learn it: $(- p, q)$
Quadratic formula, $a x^{2} + b x + c = 0$ On the formula sheet: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

On the formula sheet = printed on the formulae sheet in the 4MA1 Higher papers, so know where to find it. Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Completing the Square worked examples

Worked example 1: Completing the square

Write $x^{2} + 14 x + 20$ in the form $(x + p)^{2} + q$ .

Solution

Halve 14: $(x + 7)^{2} = x^{2} + 14 x + 49$ .
$x^{2} + 14 x + 20 = (x + 7)^{2} - 49 + 20$ .
$= (x + 7)^{2} - 29$ .

4 steps

Tip: The turning point of

y = (x + 7)^{2} - 29

(- 7, - 29)

Worked example 2: Completing the square with a ≠ 1

Write $3 x^{2} + 12 x - 5$ in the form $a (x + b)^{2} + c$ .

Solution

Factor 3 from the $x$ terms: $3 (x^{2} + 4 x) - 5$ .
$x^{2} + 4 x = (x + 2)^{2} - 4$ .
$3 (x + 2)^{2} - 12 - 5 = 3 (x + 2)^{2} - 17$ .

4 steps

Tip: Remember to multiply the −4 by 3.

Worked example 3: Minimum value of a quadratic · 3 marks

Find the minimum value of $x^{2} + 3 x + 5$ and the value of $x$ at which it occurs.

Mark scheme

M1 (x + 3/2)² seen
A1 (x + 3/2)² + 11/4
A1 minimum 11/4 when x = −3/2

Answer: Minimum value $\frac{11}{4}$ when $x = - \frac{3}{2}$

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Completing the Square exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

Write $x^{2} + 6 x + 11$ in the form $(x + p)^{2} + q$ .

Show the answer and mark scheme

M1 (x + 3)² seen
A1 (x + 3)² + 2

Answer: $(x + 3)^{2} + 2$

Question 2 · easy · 2 marks

Write $x^{2} - 10 x + 4$ in the form $(x + p)^{2} + q$ .

Show the answer and mark scheme

M1 (x − 5)² seen
A1 (x − 5)² − 21

Answer: $(x - 5)^{2} - 21$

Question 3 · easy · 1 mark

Write down the coordinates of the turning point of the graph of $y = (x - 4)^{2} + 7$ .

Show the answer and mark scheme

B1 (4, 7)

Answer: $(4, 7)$

Question 4 · medium · 2 marks

Write $x^{2} + 5 x - 2$ in the form $(x + p)^{2} + q$ .

Full worked solution Free, no sign-in

M1 (x + 5/2)² seen
A1 (x + 5/2)² − 33/4 oe

Answer: ${(x + \frac{5}{2})}^{2} - \frac{33}{4}$

Every question here has a full worked solution. See them all →

Question 5 · medium · 3 marks

By completing the square, solve $x^{2} + 8 x + 3 = 0$ . Give your answers in the form $a \pm \sqrt{b}$ .

Show the answer and mark scheme

M1 (x + 4)² − 16 + 3 = 0
M1 (x + 4)² = 13 or x + 4 = ±√13
A1 −4 ± √13

Answer: $x = - 4 \pm \sqrt{13}$

Question 6 · medium · 3 marks

By writing $x^{2} - 12 x + 30$ in the form $(x + p)^{2} + q$ , find the coordinates of the turning point of the curve $y = x^{2} - 12 x + 30$ .

Show the answer and mark scheme

M1 (x − 6)² seen
A1 (x − 6)² − 6
B1 (6, −6) ft their p and q

Answer: $(x - 6)^{2} - 6$ ; turning point $(6, - 6)$

Question 7 · hard · 3 marks

Solve $x^{2} - 6 x - 11 = 0$ by completing the square. Give your answers in the form $a \pm b \sqrt{5}$ , where $a$ and $b$ are integers.

Show the answer and mark scheme

M1 (x − 3)² − 9 − 11 = 0
M1 x − 3 = ±√20
A1 3 ± 2√5

Answer: $x = 3 \pm 2 \sqrt{5}$

Common completing the square mistakes

Forgetting to subtract the square. $x^{2} + 8 x + 3$ is not $(x + 4)^{2} + 3$ : $(x + 4)^{2}$ already contains $+ 16$ , so take it away.
The wrong sign in the turning point. $(x + 4)^{2} - 13$ has its turning point at $x = - 4$ , not $x = 4$ : the bracket is zero when $x = - 4$ .
Not multiplying back. In $2 [(x + 3)^{2} - 9] + 5$ the $- 9$ is multiplied by 2 as well, giving $- 18 + 5 = - 13$ .
Losing the ±. $(x + 4)^{2} = 13$ gives $x + 4 = \sqrt{13}$ or $x + 4 = - \sqrt{13}$ : there are two solutions.

Where students lose marks across 4MA1, from the examiner reports →

Completing the Square FAQ

What is completing the square used for?

Three jobs in 4MA1: writing a quadratic in the form (x + p)² + q, finding the turning point (and so the minimum value) of a quadratic graph, and solving a quadratic equation exactly.

How do I find the turning point from completed square form?

For y = (x + p)² + q the turning point is (−p, q): change the sign of the number in the bracket and keep the sign of the number outside. For y = (x − 5)² + 2 it is (5, 2), a minimum.

Can I use a calculator for completing the square in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Completing the Square revision notes · 4MA1 formula sheet

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Completing the Square: exam questions and answers, notes and worked examples

Try it first

Completing the Square notes

When the coefficient of x² is 1

When the coefficient of x² is not 1

Turning points

Solving by completing the square

Key formulas

Completing the Square worked examples

Worked example 1: Completing the square

Solution

Worked example 2: Completing the square with a ≠ 1

Solution

Worked example 3: Minimum value of a quadratic · 3 marks

Mark scheme

Completing the Square exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 2 marks

Question 3 · easy · 1 mark

Question 4 · medium · 2 marks

Full worked solution Free, no sign-in

Question 5 · medium · 3 marks

Question 6 · medium · 3 marks

Question 7 · hard · 3 marks

Common completing the square mistakes

Completing the Square FAQ

Next steps

Practise completing the square with your progress saved

More Unit 3 topics: Equations & Sequences