A-level · Beta

Quadratic inequalities

Connect a parabola with exact solution intervals, including repeated roots.

Choose an inequality, then change the coefficients. Green on the number line marks the solution.

f(x)=3−4x+x2f(x)=3-4x+x^{2}
Choose an inequality, then change the coefficients. Green on the number line marks the solution.-4-4-1.75-20.502.75254xy

Solve f(x) < 0. Read the solution set below.

Discriminant b² − 4ac
4
Turning point (approx.)
(2, -1)
Real roots (approx.)
1, 3

Solution set: (1,3)(1,3)

13

Filled endpoints are included; open endpoints are excluded. Infinite intervals continue beyond the displayed number line.

The red point is the turning point. Graph heights are clipped for readability; root labels are rounded, while the solution set uses exact values.

Predict, test and explain

Step 1 of 3

Predict

Is x² − 4x + 3 < 0 true outside the roots?

InvestigateWhy do strict inequalities exclude the roots?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Is x² − 4x + 3 < 0 true outside the roots?

−x2+4x−3≥0-x^2+4x-3\ge0

Hint

Find the roots, then test the sign on each interval.

Reveal answer

The upward-opening parabola lies below the axis between its two roots. Strict inequalities exclude both roots.

1≤x≤31\le x\le3

Build an explanation

Hint 1

Find the roots, then test the sign on each interval.

Hint 2

The upward-opening parabola lies below the axis between its two roots. Strict inequalities exclude both roots.

Hint 3

Use the model to check a prediction, then explain the result.

Worked example

This example uses fixed values, separate from the diagram controls.

Solve −x² + 4x − 3 ≥ 0.

  1. Factorise as −(x − 1)(x − 3). The roots are 1 and 3.
  2. The parabola opens downwards and is positive between its roots.
  3. Include the endpoints because equality is allowed.

Answer: 1 ≤ x ≤ 3.

Connect this idea

Watch out: The upward-opening parabola lies below the axis between its two roots. Strict inequalities exclude both roots.

Specification and learning route

AQA B5 · Edexcel Pure 2.5

AS and A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Factorising quadratics; Real roots; Inequality symbols.

AQA specification · Edexcel specification