AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 10

Mark scheme

Show the mark scheme

Q Answer Mark Comments

x2 + 2x – 5x – 10 oe with brackets expanded

four terms in any order with three correct

from x2 (+)2x –5x –10

M1 terms may be seen in a grid

implied by x2 – 3x + k (k ≠ 0)

or ax2 – 3x – 10 (a ≠ 0)

x2 – 3x – 130 (= 0) oe expression/equation with brackets

M1dep expanded

eg x2 + 2x – 5x – 10 = 120

For their three-term quadratic, do not accept x2 – 3x – 10 (= 0) as their

three-term quadratic

correctly factorises eg (x + 10)(x – 13) (= 0)

10 − − 3 ± ( 3)−2 − 4 ×1× −130

or correctly substitutes into the

eg

quadratic formula M1 2×1

or correctly completes the square 3

eg (x =) 1.5 ± +130

to the form x = … for their quadratic 2

or

–10 and 13

13 A1 SC1 31.5 oe

Additional Guidance

The first and third marks may be awarded for correct work with no answer

or incorrect answer, even if this is seen amongst multiple attempts

SC1 is for using the perimeter

Trial and improvement is 0, 3 (for –10 and 13) or 4 marks

How to answer it

Setting Up and Solving Geometric Quadratics

📌 What this question tests

This question tests your ability to form an algebraic equation from a 2D geometric context (the area of a rectangle), expand double brackets, rearrange the terms into the standard quadratic form ax² + bx + c = 0 , solve the quadratic equation using factorisation (or the quadratic formula), and choose the correct contextual solution for length.

Question 10: Finding the Dimensions of a Rectangle [4 Marks]

A rectangle has sides (x + 2) cm and (x − 5) cm. Its area is 120 cm². Find the value of x.

📐 Step-by-Step Solution

1 Set up the area formula:
Area = length × width
(x + 2)(x − 5) = 120

2 Expand brackets (FOIL / Grid):
x² − 5x + 2x − 10 = 120
x² − 3x − 10 = 120

3 Rearrange to equal 0:
Subtract 120 from both sides:
x² − 3x − 130 = 0

4 Factorise the quadratic:
Find two numbers that multiply to −130 and add to −3 (these are +10 and −13):
(x + 10)(x − 13) = 0

5 Solve and choose contextually valid root:
x = −10 or x = 13
Since length must be positive: if x = −10 , side (x − 5) = −15 (impossible).
Therefore, x = 13 .

✅ Mark Scheme Breakdown

M1: Correct expansion of brackets with at least 3 correct terms:
x² + 2x − 5x − 10 (or implied by x² − 3x + k ).
M1 (dep): Correctly forming the 3-term quadratic equal to zero:
x² − 3x − 130 = 0 .
M1: Correct method to solve their 3-term quadratic (e.g. factorising to (x + 10)(x − 13) , substitution into the quadratic formula, or completing the square).
A1: Final answer of x = 13 only (must reject −10).

Special Case: A student who sets up an equation using perimeter instead of area can gain at most SC1 for 31.5 .

💡 Key Knowledge

Standard Quadratic Form: You cannot factorise or solve a quadratic equation until all terms are on one side and it equals zero: ax² + bx + c = 0 .
Finding Factor Pairs: For c = −130 , list the factor pairs: (1, 130), (2, 65), (5, 26), (10, 13). Notice that 10 and 13 have a difference of 3.
Alternative Method (Quadratic Formula):
x = (−b ± √(b² − 4ac)) / 2a
x = (−(−3) ± √((−3)² − 4(1)(−130))) / 2(1)
x = (3 ± √(9 + 520)) / 2 = (3 ± √529) / 2 = (3 ± 23) / 2
Giving x = 13 or x = −10 .

🧠 Exam Technique

Contextual Sense-Check: A length cannot be negative! Always write down both mathematical solutions ( x = 13 and x = −10 ), but clearly underline or state x = 13 on the final answer line. Leaving both values on the answer line loses the final mark.
Double-check bracket signs: Multiplying (+2) × (−5) gives −10 , not +10 . Sign slips here ruin the whole quadratic.

❌ Common Pitfalls & Traps to Avoid

Forgetting to subtract the area: A very common error is factorising x² − 3x − 10 = 0 into (x − 5)(x + 2) and ignoring the 120 entirely. The mark scheme specifically states: "do not accept x² − 3x − 10 = 0 as their three-term quadratic".
Confusing Area with Perimeter: Writing 2(x + 2) + 2(x − 5) = 120 gives 4x − 6 = 120 ⇒ x = 31.5 . This only earns 1 special case mark out of 4!
Sign error when squaring negative numbers in the formula: Calculating (−3)² as −9 instead of +9 leads to taking the square root of an incorrect number.

Topics

Algebra · Geometry and measures · 3.2.1 Notation, vocabulary and manipulation · 3.2.3 Solving equations and inequalities · 3.4.2 Mensuration and calculation