Watch out
Using b − 4ac instead of b² − 4ac
Scottish National 5 Mathematics
National 5 Maths Discriminant
Use the discriminant to calculate and describe the number and nature of real roots.
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.
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
Check
Check roots by substitution and compare the turning point or discriminant conclusion with the equation's graph.
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.
Worked examples
Worked example 1
Two distinct real roots
Use the discriminant for x² − 5x + 6 = 0.
- a = 1, b = −5, c = 6.
- b² − 4ac = (-5)² − 4(1)(6)
- 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.
- a = 1, b = −6, c = 9.
- b² − 4ac = (-6)² − 4(1)(9)
- 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.
- a = 2, b = 1, c = 3.
- b² − 4ac = 1² − 4(2)(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.