Quadratic equations (formula)

The quadratic formula solves any quadratic equation ax² + bx + c = 0, including those that cannot be factorised. It is given on the formula sheet in some exams.

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Key facts to remember

1The formula is x = (−b ± √(b² − 4ac)) / (2a).
2The discriminant b² − 4ac determines the number of solutions: > 0 → two solutions, = 0 → one solution, < 0 → no real solutions.
3Substitute a, b and c carefully — include signs.
4Give answers to the required degree of accuracy (usually 3 significant figures or 2 decimal places).

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Formulas

Quadratic formula

x = (−b ± √(b² − 4ac)) / (2a)

For ax² + bx + c = 0

Discriminant

Δ = b² − 4ac

> 0: two roots, = 0: one root, < 0: no real roots

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Worked examples

Example 1

Solve 2x² − 5x − 3 = 0 using the quadratic formula. Give answers to 2 d.p.

Working

a = 2, b = −5, c = −3
Discriminant: (−5)² − 4(2)(−3) = 25 + 24 = 49
x = (5 ± √49) / 4 = (5 ± 7) / 4
x = (5 + 7)/4 = 3 or x = (5 − 7)/4 = −0.5

Answerx = 3 or x = −0.5

Example 2

Solve x² + 3x − 1 = 0. Give answers to 2 d.p.

Working

a = 1, b = 3, c = −1
Discriminant: 9 + 4 = 13
x = (−3 ± √13) / 2
x = (−3 + 3.606) / 2 ≈ 0.30 or x = (−3 − 3.606) / 2 ≈ −3.30

Answerx ≈ 0.30 or x ≈ −3.30

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Common mistakes

✗Using b instead of −b (forgetting to negate b in the numerator).

✗Calculating b² − 4ac as (b² − 4a)c or similar — apply BIDMAS carefully.

✗Forgetting the ± and giving only one solution.

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Exam tips

✓Write a, b and c clearly before substituting — include their signs.

✓Compute the discriminant first, then take the square root, to reduce errors.

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