Completing the square is a key algebra technique you'll encounter in your GCSE Maths course, particularly if you're studying for the Higher Tier. It's a method for rewriting a quadratic expression (like (x² + 6x - 7)) in a very specific and useful form: ((x + a)² ± b), where (a) and (b) are integers.
This form isn't just an exam trick. It helps you:
- Find the minimum or maximum point (the vertex) of a quadratic graph.
- Solve quadratic equations when factorising is tricky.
- Sketch quadratic graphs more accurately.
Let's break down the exam question and the method, step-by-step.
The Question: Write (x² + 6x - 7) in the form ((x + a)² - b)
Our goal is to manipulate the expression (x² + 6x - 7) so it looks like a squared bracket, plus or minus a constant number.
Step 1: Focus on the (x²) and (x) terms
First, we ignore the constant term (the -7) for a moment. We only look at:
x² + 6x
We want to make this part of the expression look like a perfect square. A perfect square expands like this:
(x + a)² = x² + 2ax + a²
Notice the coefficient of (x) in the expansion is (2a). In our expression, the coefficient of (x) is 6.
Key step: Compare (2a) with 6.
2a = 6
a = 3
So, if (a = 3), then ((x + 3)²) would expand to (x² + 6x + 9).
Step 2: Create the perfect square and adjust
We've worked out that ((x + 3)² = x² + 6x + 9).
But our original expression is (x² + 6x - 7), not (x² + 6x + 9).
Let's write this as an equation to see the difference:
x² + 6x - 7 = (x + 3)² ?
We know ((x+3)²) gives us an extra +9. To get from +9 down to our original -7, we need to subtract 16.
Why 16? Because (9 - 16 = -7).
So we can write:
x² + 6x - 7 = (x + 3)² - 16
Let's check by expanding the right-hand side:
(x+3)² - 16 = (x² + 6x + 9) - 16 = x² + 6x - 7
Perfect!
The Completed Solution
x² + 6x - 7 = (x+3)² - 16
- Here, (a =
- 3) and (b = 16), both integers as the question required.
Visualising the Process: The 'Area Model'
Completing the square can be thought of as literally trying to build a square from the (x²) and (x) terms. The diagram below shows why we needed to add and then subtract the 9.
flowchart TD
A["Start: x² + 6x - 7"] --> B["Focus on x² + 6x"]
B --> C["To make x² + 6x a perfect square,, we need to add 9, because (x+3)² = x² + 6x + 9"]
C --> D["So, x² + 6x = (x+3)² - 9"]
D --> E["Now include the original -7:, x² + 6x - 7 = (x+3)² - 9 - 7"]
E --> F["Final Answer: (x+3)² - 16"]
Your Turn: The General Method
For any quadratic in the form (x² + bx + c), follow this recipe:
Identify (b): Look at the coefficient of (x) (in our example, (b=6)).
Find (a): Halve this coefficient. (a = b ÷ 2). (Here, (6 ÷ 2 = 3)).
Write the square: ((x + a)²) expands to (x² + bx + a²).
Adjust for the constant: You've added (a²) (here, 9). To get back to the original expression, you need to subtract it and then add the original constant (c).
- In one step: (x² + bx + c = (x + a)² - a² + c).
Formula Alert: You might see this written as a formula:
x² + bx + c = (x + b2)² - (b2)² + c
Why This is a Common GCSE Question
Exam boards like AQA, Edexcel, and OCR frequently test this skill because it combines several core algebraic concepts: expanding brackets, factorising, and rearranging formulae. It's often worth 2-3 marks. Marks are typically awarded for:
- Correctly identifying the value of (a) (1 mark).
- Correctly finding the value of (b) and writing the final answer (1 mark).
Top Tips for Exam Success
Check your signs! This is the most common error. If your original equation has (x² - 4x), then (a) would be -2, giving ((x - 2)²).
Always expand your final answer as a quick check. It takes 10 seconds and can save you losing easy marks.
If the coefficient of (x²) is not 1 (e.g., (2x² + 8x + 5)), you must factorise that number out of the (x²) and (x) terms first. This is an extension you might see at Higher Tier.
Practice! Try these:
- (x² + 10x + 15) (Answer: ((x+5)² - 10))
- (x² - 8x + 3) (Answer: ((x-4)² - 13))
Final Thoughts
Completing the square might seem like a mysterious process at first, but it's just a reliable, step-by-step method. By focusing on the (x²) and (x) terms to build your perfect square, and then carefully adjusting the constant, you can tackle any question of this type in your GCSE exam. Remember the core idea: halve the (x) coefficient, square it, add and subtract it. Good luck with your revision!