AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 23

Mark scheme

Show the mark scheme

Q Answer Mark Comments

(x + 6)2… or (x + 6)(x + 6)… 2

accept x +

M1 2

implied by grid for (x + 6)(x + 6)

23(a) 2

(x + 6) + 14 A1 condone a = 6 and b = 14

Additional Guidance

(x + 6)2 may be (x + 6)(x + 6) or (6 + x)2 or (x + 6)(6 + x) throughout

23(b) (+)3, –8 B2 B1 for each coordinate

How to answer it

Completing the Square & Finding Turning Points

📌 What this question tests

This question assesses your ability to algebraically complete the square on a quadratic expression in the standard form x² + bx + c, and use completed-square form y = (x + p)² + q to directly state the coordinates of the turning point (vertex) of a parabolic curve without calculus.

Question 23 (a)

Write x² + 12x + 50 in the form (x + a)² + b where a and b are integers. [2 marks]

✅ Correct Answer

(x + 6)² + 14

Also accepted: a = 6 and b = 14 , or writing as (x + 6)(x + 6) + 14 .

💡 Key Knowledge

Standard method: x² + bx + c = (x + b/2)² − (b/2)² + c
Take the coefficient of x (which is 12) and halve it to find a = +6 .
Always subtract the square of this value: 6² = 36 .

📐 Step-by-Step Calculation

Halve the middle term coefficient:
Half of +12 is +6. Write the squared bracket: (x + 6)² .
Subtract the square of that number:
Expanding (x + 6)² gives x² + 12x + 36 , which has an extra 36.
So, x² + 12x = (x + 6)² − 36 .
Add the constant term:
(x + 6)² − 36 + 50 = (x + 6)² + 14 .

🧠 Exam Technique

Always expand your completed square answer back out in the margin to check:

(x + 6)² + 14 = x² + 12x + 36 + 14 = x² + 12x + 50 ✔

It takes 10 seconds and guarantees you don't drop the accuracy mark!

❌ Common Errors

Adding instead of subtracting: Writing (x + 6)² + 36 + 50 = (x + 6)² + 86 .
Forgetting to square: Subtracting 6 instead of 6² (writing 50 − 6 = 44 ).
Sign errors: Writing (x − 6)² .

Mark Scheme Breakdown:
• M1: For writing (x + 6)²... or (x + 12/2)² .
• A1: Fully correct expression (x + 6)² + 14 .

Question 23 (b)

A curve has the equation y = (x − 3)² − 8. Write down the coordinates of the turning point of the curve. [2 marks]

✅ Correct Answer

(3, −8)

Also written as x = 3, y = −8 .

💡 Key Knowledge

For any quadratic in the form y = (x − p)² + q, the turning point (vertex) is at (p, q).
A squared bracket is always ≥ 0. The minimum value of (x − 3)² is 0, which occurs when x − 3 = 0 ⇒ x = 3 .
When x = 3 , y = 0 − 8 = −8 .

🧠 Exam Technique

Change the sign inside, keep the sign outside:
Inside bracket: (x − 3) → x-coordinate is +3.
Outside bracket: − 8 → y-coordinate is −8.
The command words "Write down" indicate no complex working or calculus is needed.

❌ Common Errors

Wrong sign on x: Giving (−3, −8) by forgetting to reverse the sign inside the bracket.
Reversing the sign on y: Giving (3, 8) . The constant outside keeps its sign.
Swapping axes: Writing coordinates in the reverse order as (−8, 3) .

Mark Scheme Breakdown:
• B1: For giving the x-coordinate as 3 (or +3 ).
• B1: For giving the y-coordinate as −8 .
Total: 2 marks (independent marks for each coordinate).

Topics

Algebra · 3.2.1 Notation, vocabulary and manipulation · 3.2.2 Graphs