Completing the Square GCSE Maths: Full Method with Worked Examples

Completing the square is one of the most versatile techniques in GCSE Maths. It appears on all three main exam boards — AQA (8300), Edexcel (1MA1), and OCR (J560) — and it's almost exclusively a Higher tier topic. You'll typically see it on Paper 2 or Paper 3, either as a standalone quadratics question or as part of a coordinate geometry question asking for the vertex or minimum of a curve. It can earn you 4–6 marks in a single question, so it's well worth mastering.

What You Need to Know First

Completing the square builds on several earlier topics. Before you start, make sure you're solid on:

Expanding double brackets, especially (x + a)²
Collecting like terms
Solving equations by taking square roots (including ±)
The fact that (x + a)² = x² + 2ax + a²
Basic knowledge of quadratic graphs — what a minimum or maximum looks like

If you need a refresher on expanding (x + a)², work through a few examples before reading on.

The Core Method: Completing the Square

Every quadratic in the form x² + bx + c can be rewritten in the form (x + p)² + q. This is called completed square form. Here's how you get there.

The method:

Write (x + half the coefficient of x)²
Expand that bracket and compare it to the original expression
Adjust by adding or subtracting a constant to make them equal

Let's go step by step with numbers.

Worked Example 1 — Complete the square for x² + 6x + 2

Step 1: Take half the coefficient of x. The coefficient of x is 6, so half of that is 3. Write: (x + 3)²

Step 2: Expand (x + 3)² = x² + 6x + 9. Our original expression is x² + 6x + 2. The difference is: 9 − 2 = 7. We have 7 too many.

Step 3: Subtract 7 to compensate. (x + 3)² − 7

Check: (x + 3)² − 7 = x² + 6x + 9 − 7 = x² + 6x + 2 ✓

So x² + 6x + 2 = (x + 3)² − 7

Worked Example 2 — Complete the square for x² − 10x + 3

Step 1: Half of −10 is −5. Write: (x − 5)²

Step 2: Expand (x − 5)² = x² − 10x + 25. Our expression has + 3 at the end. We have 25 − 3 = 22 too many.

Step 3: Subtract 22. (x − 5)² − 22

Check: x² − 10x + 25 − 22 = x² − 10x + 3 ✓

So x² − 10x + 3 = (x − 5)² − 22

The shortcut formula, once you're comfortable: for x² + bx + c, the answer is always (x + b/2)² − (b/2)² + c. But understanding why it works — through the steps above — is what lets you adapt when the question changes slightly.

Finding the Minimum Point and Solving Quadratics

Once you have the completed square form, you can immediately read off two things: the minimum point of the curve, and you can solve the equation by rearranging.

Reading off the minimum point

The expression (x + p)² + q reaches its minimum when the bracket equals zero — since (x + p)² is always ≥ 0, adding q gives a minimum value of q. This happens when x = −p.

So for y = (x + 3)² − 7, the minimum point is (−3, −7). For y = (x − 5)² − 22, the minimum point is (5, −22).

Notice the sign flip: if the bracket is (x + 3), the x-coordinate of the minimum is x = −3.

Worked Example 3 — Solve x² + 6x + 2 = 0, giving answers in surd form

We already know x² + 6x + 2 = (x + 3)² − 7.

Set equal to zero: (x + 3)² − 7 = 0 (x + 3)² = 7 x + 3 = ±√7 x = −3 ± √7

So x = −3 + √7 or x = −3 − √7.

These are exact surd answers — exactly what the mark scheme wants when a question says "leave your answer in surd form."

What about when the coefficient of x² isn't 1?

If the quadratic is 2x² + 8x + 3, you first factorise out the coefficient of x²: 2(x² + 4x) + 3

Now complete the square on the bracket: x² + 4x → (x + 2)² − 4

So: 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5

This version appears more often on AQA and Edexcel Higher papers in recent years, so make sure you're comfortable with it.

Worked Example 4 — Complete the square for 3x² − 12x + 7

Step 1: Factorise the 3 from the x terms. 3(x² − 4x) + 7

Step 2: Complete the square inside the bracket. Half of −4 is −2. (x − 2)² = x² − 4x + 4, so x² − 4x = (x − 2)² − 4. Replace: 3[(x − 2)² − 4] + 7

Step 3: Expand the outer bracket. 3(x − 2)² − 12 + 7 = 3(x − 2)² − 5

Minimum point: x = 2, y = −5, so minimum is (2, −5).

Common Mistakes Students Make

Getting the sign wrong on p. From (x + 3)² − 7, students often write the minimum as (3, −7) instead of (−3, −7). The x-coordinate of the minimum is −p, not p. Say it out loud: "the bracket is zero when x is the opposite of what's in the bracket."
Forgetting the ± when taking square roots. When solving (x + 3)² = 7, writing x + 3 = √7 gives only one solution. Every quadratic equation has two roots (unless it's a perfect square). Always write ±.
Errors with fractions when b is odd. If the coefficient of x is 5, then b/2 = 5/2 = 2.5 and (b/2)² = 6.25. Students often round or get muddled. Write fractions explicitly: (5/2)² = 25/4.
Not adjusting after factorising the leading coefficient. In 2(x² + 4x) + 3, after completing the square inside to get 2[(x+2)² − 4] + 3, students forget to multiply the −4 by 2, writing 2(x+2)² − 4 + 3 instead of 2(x+2)² − 8 + 3.
Confusing minimum with maximum. If the coefficient of x² is negative (e.g. −x² + …), the parabola opens downward and has a maximum, not a minimum. The vertex is still read off in the same way, but describe it correctly.

Exam Tips for Completing the Square

The method mark comes from the structure. AQA and Edexcel mark schemes typically award an M1 for writing (x ± p)² even if p is wrong, as long as the method is clearly shown. Never skip the intermediate lines.
Write out your check. Expand your completed square form back to x² + bx + c as a separate line labelled "check." Examiners love it, and it costs you nothing if correct — but catches errors before they cost you marks.
For "minimum point" questions, write coordinates clearly. Don't just write "x = −3, y = −7" — write it as a coordinate pair (−3, −7) or state "the minimum point is (−3, −7)" explicitly.
Surd solutions from completing the square are exact. If the question says "give exact answers" or "leave in surd form," completing the square is the right method — the quadratic formula will work too, but completing the square with neat numbers produces cleaner answers.
Watch for combined topics. Completing the square often appears in questions about quadratic graphs — finding the turning point, the axis of symmetry (x = −p), or the y-intercept. Read the whole question before choosing your method.

Practise Now

The best way to get confident with completing the square is to practise exam-style questions with instant feedback. Try completing the square questions now on GCSEMathsAI — AI-generated questions matched to your board and tier, with detailed marking and personalised feedback.