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 Video by Zeeshan Zamurred Quadratics

Edexcel AS Level Maths: 2.5 The DiscriminantWatch the full walkthrough 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.

Test yourself

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