2.5PureCore

The Discriminant

The discriminant is the part of the quadratic formula under the square root, $b^{2} - 4 a c$ . Its sign alone tells you how many real roots a quadratic has — without solving the equation at all.

25 min Quadratics

Edexcel AS Level Maths: 2.5 The DiscriminantCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Calculate the discriminant of a quadratic
Use the discriminant to count real roots
Link the discriminant to where the curve meets the x-axis
Solve problems where the number of roots is given

What the discriminant is

For $a x^{2} + b x + c = 0$ , the discriminant is $b^{2} - 4 a c$ . Because it lives under the square root in the quadratic formula, its sign decides whether that root is real, zero or impossible.

Δ = b^{2} - 4 a c

Read off

a

b

and

c

from standard form, then evaluate.

Counting the roots

There are three cases. A positive discriminant means two different real roots; zero means one repeated root; negative means no real roots at all.

b^{2} - 4 a c > 0 \Rightarrow two distinct real roots

The curve crosses the x-axis twice.

b^{2} - 4 a c = 0 \Rightarrow one repeated root

The curve touches the x-axis (tangent).

b^{2} - 4 a c < 0 \Rightarrow no real roots

The curve never meets the x-axis.

a = 1

b = 3

c = 5

b^{2} - 4 a c = 9 - 20 = - 11 < 0

Answerno real roots

Finding unknowns

A very common exam style gives a condition on the roots (e.g. “equal roots”) and asks for an unknown constant. Translate the condition into a discriminant statement, then solve.

1Equal roots means

b^{2} - 4 a c = 0

2Here

b = k

, so

k^{2} - 4 (1) (9) = 0

, i.e.

k^{2} = 36

Answer

k = \pm 6

Tip — “Equal roots”, “repeated root” and “touches the x-axis” all mean the discriminant is exactly 0.

Formula recap

Δ = b^{2} - 4 a c

The discriminant.

Δ > 0

Two distinct real roots.

Δ = 0

One repeated root (tangent to x-axis).

Δ < 0

No real roots.

Common mistakes to avoid

Forgetting the minus sign, e.g. computing b² + 4ac.

It is b² − 4ac. Keep the subtraction.

Saying a negative discriminant means “two negative roots”.

A negative discriminant means NO real roots at all.

Key takeaways

The discriminant is Δ = b² − 4ac.
Δ > 0: two roots; Δ = 0: one repeated root; Δ < 0: none.
Δ = 0 is the “equal roots” / “tangent to the x-axis” condition.
For unknown-constant problems, set up the right discriminant condition first.

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Real past-paper questions on The Discriminant, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.

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Exam questions

3 real exam questions on The Discriminant

9 marks · 3 papers — pulled from past papers and marked mark‑by‑mark, so what you miss feeds straight into your weak‑topic list.

Practise these

The Discriminant

What the discriminant is

Counting the roots

Finding unknowns

Practise this topic

3 real exam questions on The Discriminant

Related topics

The Discriminant

What the discriminant is

Counting the roots

Finding unknowns

Practise this topic

3 real exam questions on The Discriminant

Related topics