Knowledge Organiser: Quadratic Inequalities
Part of Quadratic Inequalities · GCSE GCSE Mathematics revision
This topic summary covers Knowledge Organiser: Quadratic Inequalities within Quadratic Inequalities for GCSE Mathematics. Revise Quadratic Inequalities in Algebra for GCSE Mathematics with 10 exam-style questions and 2 flashcards. This topic appears less often, but it can still be a useful differentiator on mixed-topic papers. It is section 5 of 5 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Topic position
Section 5 of 5
Practice
10 questions
Recall
2 flashcards
Knowledge Organiser: Quadratic Inequalities
Key Terms
- Quadratic inequality: An inequality involving x² (e.g. x² − 5x + 6 < 0)
- Critical values / roots: The x-values where the quadratic equals zero
- U-shaped parabola: Graph of ax² + bx + c when a > 0 — positive outside the roots
- Inside the roots: The region between the two roots where a U-shape is below the x-axis
- Outside the roots: The two separate regions beyond the roots where the U-shape is above the x-axis
Must-Know Facts
- Step 1: rearrange to ax² + bx + c > 0 (or < 0) with 0 on one side
- Step 2: find the roots by solving ax² + bx + c = 0
- Step 3: sketch the parabola (U-shape if a > 0)
- For ax² + bx + c < 0 (a > 0): solution is BETWEEN the roots
- For ax² + bx + c > 0 (a > 0): solution is OUTSIDE the roots (two separate regions)
- "Between" = one inequality; "outside" = two separate inequalities connected by OR
Key Methods
- Rearrange → solve = 0 → find roots p and q (p < q) → sketch → read from graph
- f(x) < 0 (U-shape): p < x < q (between the roots)
- f(x) > 0 (U-shape): x < p OR x > q (outside the roots)
- Write "OR" (not "AND") when the solution is two separate regions
Key Formulas
- Solve ax² + bx + c < 0: find roots p and q, sketch, read solution from graph
- U-shape (a>0): below x-axis between roots → p < x < q
- U-shape (a>0): above x-axis outside roots → x < p OR x > q
- ∩-shape (a<0): reverse the above regions
Common Mistakes
- Writing "AND" instead of "OR": For the outside region (two separate parts), use OR — "x < p OR x > q" not "x < p AND x > q"
- Not sketching: Always sketch the parabola to visualise which region satisfies the inequality
- Including boundary with strict inequality: For < or >, use open circles (boundary not included); for ≤ or ≥, use closed circles
- Negative a: The parabola opens downward — the "inside" and "outside" regions are swapped
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Practice Questions for Quadratic Inequalities
For a positive (U-shaped) parabola with roots at x = 2 and x = 5, the expression is negative (below the x-axis) for which values of x?
Explain what the solution set of x² + 2x + 5 > 0 is, justifying your answer using the discriminant.
Quick Recall Flashcards
10 questions on Quadratic Inequalities — practise free
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