AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 22

Solving Quadratic Inequalities

🎯 Target Grade: 8 / 9

What this question tests

This question assesses your ability to manipulate and solve a non-linear inequality involving a quadratic expression where the coefficient of x² is greater than 1:

Rearranging all terms onto one side to form a quadratic inequality compared with zero ( ax² + bx + c > 0 ).
Finding critical values (roots) using factorisation, completing the square, or the quadratic formula.
Interpreting the inequality sign geometrically (identifying the correct regions above or below the x-axis).
Expressing distinct, disconnected solution sets using separate inequalities rather than invalid combined notation.

Question 22 (4 Marks)

Solve: 2x² > 12 − 5x

📐 Step-by-Step Calculation

1 Rearrange to standard form:
Add 5x and subtract 12 from both sides:
2x² + 5x − 12 > 0
2 Find critical values (solve = 0):
Factorise 2x² + 5x − 12 = 0 :
Need two numbers that multiply to 2 × (−12) = −24 and add to +5 . These are +8 and −3 .
Rewrite middle term: 2x² + 8x − 3x − 12 = 0
2x(x + 4) − 3(x + 4) = 0
(2x − 3)(x + 4) = 0
Roots: x = 1.5 (or 3/2 ) and x = −4
3 Identify the required region:
The quadratic has a positive x² coefficient (a U-shaped parabola). We want where the curve is strictly above the x-axis ( > 0 ).
This occurs in the two outer tails: to the left of the lower root and to the right of the upper root.
4 Write the final inequalities:
x < −4 or x > 1.5

✅ Final Answer & Mark Breakdown

Answer: x > 1.5 and x < −4 (or x > 3/2 , separated by a comma, 'or', or 'and')

Mark Scheme Breakdown:

B1: Rearranging to 2x² + 5x − 12 (> 0) or 12 − 5x − 2x² (< 0) . Can be implied by finding roots 1.5 and −4.
M1: Valid method to solve their 3-term quadratic (e.g. correct factorisation: (2x − 3)(x + 4) , completing the square, or formula).
A1ft: Both critical values correct: 1.5 and −4 (follow-through from their 3-term quadratic).
B1ft: Final answer correctly stating both disjoint inequalities: x > 1.5 and x < −4 .

💡 Key Knowledge: Sketching the Curve

Never try to deduce quadratic inequality regions purely in your head. Always sketch a quick quadratic curve:

Examiner Sketch Visualisation:
Draw an upright U-shaped curve cutting the horizontal x-axis at two points: −4 (on the left) and 1.5 (on the right).
• For > 0 : Shade the regions above the horizontal axis (left of −4, right of 1.5).
• For < 0 : Shade the region below the horizontal axis (between −4 and 1.5).

Since the question specifies > 0 , choose the outer regions where the curve rises above the axis.

🧠 Exam Technique: Testing Points

To double check your inequality boundaries during the exam, test a simple value in each region:

Test x = 0 (between −4 and 1.5):
2(0)² + 5(0) − 12 = −12 .
Is −12 > 0 ? False. Therefore, the middle region is NOT part of the solution.
Test x = 2 (in region x > 1.5):
2(2)² + 5(2) − 12 = 8 + 10 − 12 = 6 .
Is 6 > 0 ? True. Confirms x > 1.5 is correct!

❌ Common Errors & Examiner Warnings

Joining disconnected inequalities into a single line:
Writing 1.5 < x < −4 or −4 > x > 1.5 is mathematically impossible (it would imply 1.5 < −4 ).
Examiner penalty: An incorrectly joined inequality instantly scores B0 for the final mark. Always keep two separate statements: x < −4 and x > 1.5 .
Reversing the inequality signs:
Students frequently write −4 < x < 1.5 because they confuse the regions for > 0 with those for < 0 .
Treating inequalities like linear equations:
Writing x(2x + 5) > 12 and attempting to conclude x > 12 or 2x + 5 > 12 is completely invalid. One side must be zero.
Sign errors when rearranging:
Moving terms across incorrectly, e.g. writing 2x² − 5x − 12 > 0 instead of +5x . Check your signs carefully when moving terms across the inequality sign.

AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 22

Question

Mark scheme

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