A Level Maths: Quadratic Inequalities and The Discriminant Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
- Inequalities (OCR)
- Inequalities (Edexcel)
- The Discriminant (OCR)
- The Discriminant (Edexcel)
- Graphing Inequalities (OCR)
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Quadratic Equations — solving and sketching quadratics is the immediate prerequisite
- Indices — needed for manipulating expressions before solving
- Straight Line Geometry — required for line-curve intersection problems
A Level Maths quadratic inequalities and the discriminant extends Quadratic Equations in two directions. First, instead of asking when $ax^2 + bx + c$ equals zero, inequalities ask when the expression is greater than or less than zero — which means knowing where the parabola sits above or below the $x$-axis. Second, the discriminant $b^2 – 4ac$ tells you how many real roots a quadratic has without you having to solve it.
To solve a quadratic inequality, you sketch the parabola, find its roots, and read off the regions where the curve is on the correct side of the axis. The shape of the curve (opening upwards if $a > 0$, downwards if $a < 0$) matters as much as the roots. You express solutions using inequality notation, set-builder notation, or interval notation depending on what the question asks. The discriminant classifies quadratics into three cases: $b^2 – 4ac > 0$ gives two distinct real roots, $b^2 – 4ac = 0$ gives one repeated root (the curve touches the $x$-axis), and $b^2 – 4ac < 0$ gives no real roots (the curve does not cross the axis). This is used heavily in problems involving tangents, equal roots, and conditions for a line to intersect a curve.
Quadratic inequalities and the discriminant are part of the Pure Maths strand of A Level Maths and connect to topics across the rest of the course for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: flip the inequality the right way when dividing or multiplying by a negative; for a parabola opening upwards, the curve is BELOW the $x$-axis between its roots, not outside them; '$x > 3$ and $x < -2$' is empty — you mean '$x > 3$ OR $x < -2$'; compute the discriminant with the correct signs after rearranging into standard form; and for the line-meets-curve condition, the discriminant condition applies to the quadratic that results from substituting one equation into the other, not to the original equation.