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.