A-level · Beta

Quadratics and the discriminant

Connect coefficients, roots and the turning point; investigate a repeated root.

Change a, b and c. Compare the discriminant with the roots and turning point.

f(x)=3−4x+x2f(x)=3-4x+x^{2}
Change a, b and c. Compare the discriminant with the roots and turning point.-4-4-1.75-20.502.75254xy

Positive discriminant: two distinct real roots.

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

Exact real roots: 11 and 33

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

Does a negative discriminant mean the quadratic is always negative?

InvestigateKeep a = 1 and b = −4. Which c makes a repeated root?

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.

Does a negative discriminant mean the quadratic is always negative?

b2−4ac>0b^2-4ac>0

Hint

Compute b² − 4ac before taking a square root.

Reveal answer

The sign of a determines which side of the axis the curve lies on when there are no real roots.

k<4k<4

Build an explanation

Hint 1

Compute b² − 4ac before taking a square root.

Hint 2

The sign of a determines which side of the axis the curve lies on when there are no real 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.

Find the values of k for which x² − 4x + k = 0 has two distinct real roots.

  1. The discriminant is (−4)² − 4(1)k = 16 − 4k.
  2. Two distinct real roots require 16 − 4k > 0.
  3. Rearrange to obtain k < 4. At k = 4 there is a repeated root.

Answer: k < 4.

Connect this idea

Watch out: The sign of a determines which side of the axis the curve lies on when there are no real roots.

Specification and learning route

AQA B3 · Edexcel Pure 2.3

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

Useful starting knowledge: Expanding brackets; Quadratic formula; Completing the square.

AQA specification · Edexcel specification