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SQA National 5 Mathematics

Discriminant

Using b² − 4ac to identify the number of real roots.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Write the quadratic in ax² + bx + c = 0 form
  • Identify a, b and c carefully, including negative signs.
  • Remember that b² means b multiplied by b, so a negative b gives a positive square.
National 5 Mathematics lesson

Explanation

The discriminant is b² − 4ac. It sits under the square root in the quadratic formula and tells you how many real roots a quadratic has.

If b² − 4ac is positive, the equation has two distinct real roots.

If b² − 4ac is zero, the equation has one repeated real root.

If b² − 4ac is negative, the equation has no real roots because the quadratic formula would need the square root of a negative number.

Key formulae and rules

  • Discriminant = b² − 4ac
  • b² − 4ac > 0: two distinct real roots
  • b² − 4ac = 0: one repeated real root
  • b² − 4ac < 0: no real roots

Watch out

Using b − 4ac instead of b² − 4ac

Check

Compare your answer with the size you expected from the question.

Exam tip

The discriminant tells you the number of real roots; it does not require you to solve the equation unless the question asks for roots.

Calculator tip

Use brackets for fractions, powers and square roots, then round only at the final line.

Worked examples

Worked example 1

Two distinct real roots

Use the discriminant for x² − 5x + 6 = 0.

  1. a = 1, b = −5, c = 6.
  2. b² − 4ac = (-5)² − 4(1)(6)
  3. 25 − 24 = 1, which is positive

Answer: The equation has two distinct real roots.

Worked example 2

One repeated real root

Use the discriminant for x² − 6x + 9 = 0.

  1. a = 1, b = −6, c = 9.
  2. b² − 4ac = (-6)² − 4(1)(9)
  3. 36 − 36 = 0

So: The equation has one repeated real root.

Worked example 3

No real roots

Use the discriminant for 2x² + x + 3 = 0.

  1. a = 2, b = 1, c = 3.
  2. b² − 4ac = 1² − 4(2)(3)
  3. 1 − 24 = −23, which is negative

Answer: The equation has no real roots.

Watch out

  • Using b − 4ac instead of b² − 4ac
  • Dropping the negative sign when b or c is negative.
  • Saying a positive discriminant gives one root instead of two distinct real roots.