AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 16

3 marks · Medium difficulty · Reasoning

Explain, without expanding brackets, why there is only one positive even value of x for which y = (x - 3)(x - 5)(x + 6) is negative.

Practise this question

Question

Question 16 states: 'x is a positive even number. y = (x - 3)(x - 5)(x + 6). Without expanding brackets, explain why there is only one value of x for which y is negative.' The question is worth 3 marks.

Mark scheme

Show the mark scheme Mark scheme for question 16 shows three B1 marks: the first B1 for showing that when x = 2, y = 24 or negative times negative times positive equals positive; the second B1 for showing that when x = 4, y = -10 or positive times negative times positive equals negative; the third B1 for showing that when x = 6, y = 36 or positive times positive times positive equals positive (or SC1 for any correct value of y for an even x > 6). Additional guidance details valid implied values and conditions for showing signs.

How to answer it

Analysing Signs in Factorised Expressions

📋 What this question tests

This question assesses reasoning and substitution without expanding brackets:

  • Condition recognition: Identifying that positive even integers start at 2, 4, 6, 8, ...
  • Sign of a product: Determining whether a product of three factors is positive or negative using directed number rules.
  • Systematic testing: Checking key critical boundary values around roots ( x = 3 and x = 5 ) to show only one valid value yields a negative output.

Question 16 Walkthrough

Explaining why only one positive even value of x produces a negative y

💡 Key Knowledge

  • Positive even numbers: x = 2, 4, 6, 8, ...
  • The roots (where each factor equals zero) are x = 3 , x = 5 , and x = -6 .
  • Since x is positive, (x + 6) is always positive.
  • Rules for signs:
    • (-) × (-) × (+) = (+)
    • (+) × (-) × (+) = (-)
    • (+) × (+) × (+) = (+)

🧠 Exam Technique

  • Obey the constraints: Do not expand the brackets. The question explicitly says "Without expanding brackets".
  • Test specific even integers: The quickest method is to test the first positive even values: 2, 4, and 6 (or any value ≥ 6).
  • State the sign clearly: Always state the resulting sign or calculate the exact numerical value of y.

📐 Step-by-Step Full Solution

Given equation: y = (x - 3)(x - 5)(x + 6) where x is a positive even number.

Step 1: Test x = 2

  • Method A (Direct calculation):
    y = (2 - 3)(2 - 5)(2 + 6) = (-1) × (-3) × (8) = 24 (which is positive)
  • Method B (Sign analysis):
    negative × negative × positive = positive
awarded 1 mark (B1) for demonstrating y is positive when x = 2

Step 2: Test x = 4

  • Method A (Direct calculation):
    y = (4 - 3)(4 - 5)(4 + 6) = (1) × (-1) × (10) = -10 (which is negative)
  • Method B (Sign analysis):
    positive × negative × positive = negative
awarded 1 mark (B1) for demonstrating y is negative when x = 4

Step 3: Test x = 6 (and beyond)

  • Method A (Direct calculation for x = 6):
    y = (6 - 3)(6 - 5)(6 + 6) = (3) × (1) × (12) = 36 (which is positive)
  • Method B (General sign argument):
    For any x ≥ 6 , all three brackets (x - 3) , (x - 5) , and (x + 6) are strictly positive, so positive × positive × positive = positive .
awarded 1 mark (B1) for demonstrating y is positive when x = 6 or for any even x > 6

Conclusion:
The only positive even number between the roots 3 and 5 is x = 4. Therefore, x = 4 is the only positive even number that gives a negative value of y .

✅ Model Answer Structure

A full mark response should clearly display all three conditions:

  • When x = 2 : y = (-1)(-3)(8) = 24 > 0 (positive)
  • When x = 4 : y = (1)(-1)(10) = -10 < 0 (negative)
  • When x = 6 : y = (3)(1)(12) = 36 > 0 (positive), and for all x > 6 , all three factors remain positive.
  • Hence, x = 4 is the only value.

❌ Common Errors to Avoid

  • Expanding brackets: Multiplying out into a cubic expression wastes significant time and contradicts the question command.
  • Only showing x = 4: Showing that x = 4 yields a negative value proves that a negative value exists, but fails to prove that it is the only one. You must test or explain the other regions ( x = 2 and x ≥ 6 ).
  • Testing odd numbers: Substituting x = 1, 3, 5 is unnecessary because the question explicitly specifies positive even numbers.
  • Missing sign conclusion: Multiplying signs but forgetting to state the final outcome (e.g. leaving (-) × (-) × (+) without stating = (+) ).

Topics

Algebra · Number · 3.2.1 Notation, vocabulary and manipulation · 3.1.1 Structure and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.