Watch out
Forgetting to put a negative b in brackets.
Back to National 5 Mathematics
SQA National 5 Mathematics
The quadratic formula
Solving quadratics when factorising is not efficient.
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
- Identify a, b and c from ax² + bx + c = 0
- Substitute negative values carefully using brackets.
Explanation
The quadratic formula solves any quadratic equation written as ax² + bx + c = 0.
It is most useful when a quadratic does not factorise neatly. Substitute a, b and c, then simplify carefully.
At National 5, method marks often come from correct substitution before calculator evaluation.
Visual support
Substitute into the quadratic formula
ax² + bx + c = 0
x = (-b ± √(b² − 4ac)) / 2a
For x² + 5x + 3 = 0: a = 1, b = 5, c = 3
Key formulae and rules
- x = (-b ± √(b² − 4ac)) / 2a
Check
Substitute each solution back into the equation. A quick check catches most sign errors.
Exam tip
Write the formula first, then a clean substitution line. This protects method marks even if arithmetic slips.
Calculator tip
Enter the numerator in brackets, especially when using a negative b or the ± answers separately.
Worked examples
Worked example 1
Use the formula
Solve x² + 5x + 3 = 0.
- Here a = 1, b = 5, c = 3
- Substitute: x = (-5 ± √(25 − 12)) / 2
- Simplify: x = (-5 ± √13) / 2.
Answer: x = (-5 + √13) / 2 or x = (-5 − √13) / 2
Worked example 2
Round calculator answers
Solve 2x² − 3x − 4 = 0, correct to 2 decimal places.
- a = 2, b = −3, c = −4.
- x = (3 ± √(9 + 32)) / 4
- x = (3 ± √41) / 4
So: x = 2.35 or x = −0.85
Worked example 3
Keep negative b under control
Solve x² − 6x + 2 = 0.
- a = 1, b = −6, c = 2.
- x = (6 ± √(36 − 8)) / 2
- x = (6 ± √28) / 2 = 3 ± √7
Answer: x = 3 + √7 or x = 3 − √7
Watch out
- Forgetting to put a negative b in brackets.
- Using 2a as 2 + a.
- Rounding too early before the final answer.