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The Discriminant

The discriminant is the part of the quadratic formula under the square root, b2−4acb^2 - 4ac. 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
1

What the discriminant is

For ax2+bx+c=0ax^2 + bx + c = 0, the discriminant is b2−4acb^2 - 4ac. Because it lives under the square root in the quadratic formula, its sign decides whether that root is real, zero or impossible.

Δ=b2−4ac\Delta = b^{2} - 4ac
Read off aa, bb and cc from standard form, then evaluate.
2

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.

b2−4ac>0⇒two distinct real rootsb^{2} - 4ac > 0 \Rightarrow \text{two distinct real roots}
The curve crosses the x-axis twice.
b2−4ac=0⇒one repeated rootb^{2} - 4ac = 0 \Rightarrow \text{one repeated root}
The curve touches the x-axis (tangent).
b2−4ac<0⇒no real rootsb^{2} - 4ac < 0 \Rightarrow \text{no real roots}
The curve never meets the x-axis.
Example 1How many real roots does x2+3x+5=0x^2 + 3x + 5 = 0 have?
1a=1a = 1, b=3b = 3, c=5c = 5.
2b2−4ac=9−20=−11<0b^2 - 4ac = 9 - 20 = -11 < 0.
Answer
no real roots
3

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.

Example 1Find kk if x2+kx+9=0x^2 + kx + 9 = 0 has equal roots.
1Equal roots means b2−4ac=0b^2 - 4ac = 0.
2Here b=kb = k, so k2−4(1)(9)=0k^2 - 4(1)(9) = 0, i.e. k2=36k^2 = 36.
Answer
k=±6k = \pm 6

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

Formula recap

Δ=b2−4ac\Delta = b^{2} - 4ac
The discriminant.
Δ>0\Delta > 0
Two distinct real roots.
Δ=0\Delta = 0
One repeated root (tangent to x-axis).
Δ<0\Delta < 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

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on The Discriminant. Use the feedback to decide what needs another look.

3 past-paper questions · 9 marks

Practise this topic

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.