Quadratic Inequalities for IGCSE Maths
Solving inequalities involving quadratic expressions. This subtopic is part of Inequalities in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding quadratic inequalities
What You Need to Know
Solving inequalities involving quadratic expressions. This subtopic is part of Inequalities in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding quadratic inequalities is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Quadratic Inequalities
Quadratic inequalities such as x² − 5x + 6 > 0 require a sketch of the parabola to identify the correct solution regions. First, solve the corresponding quadratic equation to find the roots; then use the shape of the parabola (∪ for positive x², ∩ for negative x²) to determine where the parabola is above or below the x-axis. The answer is expressed as a union of intervals or a single interval.
Step-by-Step Method
- 1
Solve the equality first
Set the expression = 0 and find the two roots (by factorising or quadratic formula). These are the boundary values.
- 2
Sketch the parabola
Mark the roots on the x-axis. For positive x² coefficient, the parabola opens upward (∪ shape). For negative, it opens downward.
- 3
Identify the required region
For > 0 (above the x-axis): on a ∪ parabola, the solution is x < smaller root OR x > larger root. For < 0 (below the x-axis): smaller root < x < larger root.
- 4
Write the solution
Write both parts clearly: e.g. x < −2 or x > 5 (for > 0 case). For < 0: just the interval e.g. −2 < x < 5.
- 5
Verify with a test value
Pick a value in each region and check it satisfies the original inequality.
Worked Example
Question
Solve x² − x − 6 < 0.
Solution
Step 1: Solve x² − x − 6 = 0. Factorise: (x − 3)(x + 2) = 0 Roots: x = 3 and x = −2. Step 2: Sketch the upward parabola (positive x²), crossing x-axis at x = −2 and x = 3. Step 3: We need < 0 (below the x-axis). On a ∪ parabola, this is BETWEEN the roots. Step 4: Solution: −2 < x < 3 Verify: Test x = 0: 0 − 0 − 6 = −6 < 0 ✓ Test x = 4: 16 − 4 − 6 = 6 > 0 (not in solution — correct).
Exam Tips for Quadratic Inequalities
- Always sketch the parabola — trying to solve quadratic inequalities without a sketch leads to wrong regions.
- For > 0 on an upward parabola: outside the roots. For < 0: between the roots. This simple rule covers most cases.
- Negative x² coefficient flips the shape — it opens downward, so the rule reverses.
- Use ≤ and ≥ (closed interval) when the original inequality includes = ; use strict < and > otherwise.
Practice Questions
Q1: Solve x² − 4 > 0.
Show hint
(x+2)(x−2) > 0. Roots at x=±2. Upward parabola, above x-axis: x < −2 or x > 2.
Q2: Solve 2x² − 7x + 3 ≤ 0.
Show hint
(2x−1)(x−3) = 0 → roots x=1/2 and x=3. Below/on the x-axis: 1/2 ≤ x ≤ 3.
Q3: Find the values of x satisfying x² + 2x > 8.
Show hint
x²+2x−8>0 → (x+4)(x−2)>0 → x<−4 or x>2.
Frequently Asked Questions
What is quadratic inequalities in IGCSE Maths?
Solving inequalities involving quadratic expressions.
Is quadratic inequalities in the Core or Extended syllabus?
Quadratic Inequalities is part of the Extended only syllabus for IGCSE Mathematics 0580.
How do I revise quadratic inequalities effectively?
Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on quadratic inequalities rather than trying to cover everything at once.
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