Completing the Square

Completing the square is a technique that rewrites a quadratic expression into a form that reveals the turning point of its graph. It is one of three methods for solving quadratic equations — alongside factorising and the quadratic formula — and it is the only one that directly shows you the minimum or maximum point of a parabola. This topic is on the Higher tier for AQA, Edexcel, and OCR, and it frequently appears in questions about sketching graphs, finding turning points, or deriving the quadratic formula. On this page you will learn the technique for both monic and non-monic quadratics, with clear examples at every stage.

What Is Completing the Square?

Completing the square rewrites a quadratic from the form ax² + bx + c into the form a(x + p)² + q.

In this new form:

The turning point of the parabola y = ax² + bx + c is at (−p, q).
If a > 0, the parabola opens upwards and (−p, q) is the minimum point.
If a < 0, the parabola opens downwards and (−p, q) is the maximum point.

x² + bx + c = (x + b/2)² − (b/2)² + c

For a monic quadratic x² + bx + c, halve the coefficient of x, square it, then adjust the constant.

The technique gets its name from literally "completing" a partial square to make a perfect square trinomial.

Step-by-Step Method

Monic quadratics (coefficient of x² is 1)

Start with x² + bx + c.
Halve the coefficient of x. If b = 6, then half is 3.
Write the perfect square bracket: (x + 3)².
Expanding (x + 3)² gives x² + 6x + 9. But you only had x² + 6x (without the +9), so you have introduced an extra 9.
Subtract the extra: (x + 3)² − 9.
Add back the original constant c: (x + 3)² − 9 + c.

Non-monic quadratics (coefficient of x² is not 1)

Factor out a from the x² and x terms: a(x² + (b/a)x) + c.
Complete the square inside the bracket using the monic method.
Multiply the adjustment by a and simplify.

Solving by completing the square

Once in the form (x + p)² = q, take the square root of both sides (remembering ±) and solve for x.

Worked Example 1 — Finding the Turning Point

Question: Write x² − 8x + 5 in the form (x + p)² + q and state the turning point of y = x² − 8x + 5.

Working:

Step 1: Halve the coefficient of x: −8 ÷ 2 = −4.

Step 2: Write the bracket: (x − 4)².

Step 3: Expanding gives x² − 8x + 16. We introduced +16 but need +5. Adjustment: 5 − 16 = −11.

Step 4: x² − 8x + 5 = (x − 4)² − 11.

Turning point: (4, −11). Since the coefficient of x² is positive, this is a minimum.

Answer: (x − 4)² − 11; turning point is (4, −11).

Worked Example 2 — Solving by Completing the Square

Question: Solve x² + 6x − 1 = 0 by completing the square. Give your answers in surd form.

Working:

Step 1: Move the constant: x² + 6x = 1.

Step 2: Halve 6 → 3. Add 3² = 9 to both sides: x² + 6x + 9 = 10.

Step 3: Write as a perfect square: (x + 3)² = 10.

Step 4: Square root both sides: x + 3 = ±√10.

Step 5: Subtract 3: x = −3 ± √10.

Check: x = −3 + √10 ≈ 0.162. Substituting: (0.162)² + 6(0.162) − 1 ≈ 0.026 + 0.974 − 1 = 0 ✓

Answer: x = −3 + √10 or x = −3 − √10

Common Mistakes

Forgetting to subtract the square you added. When you write (x + 3)², you have introduced +9. You must subtract 9 to keep the expression equivalent.
Getting the sign inside the bracket wrong. If the coefficient of x is −8, the bracket should contain −4, not +4. Halve including the sign.
Not factoring out a for non-monic quadratics. If the coefficient of x² is not 1, you must factor it out before completing the square.
Losing the ± when solving. Taking a square root produces two values. Writing only the positive root loses one solution.
Confusing the turning point coordinates. In (x + 3)² − 7, the turning point is (−3, −7), not (3, −7). The x-coordinate is the opposite sign of what appears in the bracket.

Exam Tips

Know what the question is asking. "Write in the form (x + p)² + q" means complete the square. "Solve by completing the square" means you must also set = 0 and find x.
Use completing the square to find turning points. If a question asks for the minimum value of a quadratic or the coordinates of the vertex, this is the method to use.
The answer to "find the minimum value" is q, not the full coordinate. Read carefully whether they want the minimum value (just q) or the turning point (both coordinates).
Practise non-monic examples. These are rarer but high-value questions. Being comfortable with 2(x + p)² + q format can earn you full marks where many students drop out.

Practice Questions

Q1: Write x² + 10x + 18 in the form (x + p)² + q.

Answer: (x + 5)² − 7

Q2: Find the turning point of y = x² − 4x + 9.

Answer: (x − 2)² + 5, so the turning point is (2, 5)

Q3: Solve x² − 2x − 7 = 0 by completing the square, giving answers in surd form.

Answer: (x − 1)² = 8, so x = 1 ± 2√2

Want to master completing the square with guided practice? Start revising with GCSEMathsAI — our AI tutor walks you through each step and builds your confidence for exam day.

For more on quadratic methods, see our guide on How to Solve Quadratic Equations at GCSE.

Summary

Completing the square rewrites ax² + bx + c as a(x + p)² + q.
For monic quadratics: halve the coefficient of x, square it, and adjust the constant.
The turning point of the parabola is at (−p, q).
This method can also be used to solve quadratic equations by isolating the squared bracket and taking square roots.
Always remember ± when taking square roots.
The sign inside the bracket is opposite to the x-coordinate of the turning point.
Completing the square is the foundation for deriving the quadratic formula and is essential for Higher tier graph questions.

Completing the Square –