Quadratic equations are the single most-tested topic in NSC Grade 12 Mathematics Paper 1. Question 1 of almost every November paper opens with one. If you can solve them quickly and cleanly, you bank easy marks and you free up time for the harder calculus and functions questions later in the paper.
This guide walks through every CAPS-aligned method — factorising, the quadratic formula, completing the square, and the discriminant — with full worked examples taken straight from NSC past papers.
What is a quadratic equation?
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0. The solutions (called roots) are the values of x that make the equation true. A quadratic always has 0, 1 or 2 real roots — and the discriminant tells you which.
Method 1 — Factorising (fastest when it works)
Use factorising whenever the trinomial factorises neatly over the integers. Example: solve x² − 5x − 14 = 0.
Look for two numbers that multiply to −14 and add to −5 → −7 and +2. So (x − 7)(x + 2) = 0, giving x = 7 or x = −2.
If you can't find the factors in 20 seconds, abandon factorising and switch to the quadratic formula. Don't waste exam time hunting.
Method 2 — The quadratic formula (always works)
The formula x = (−b ± √(b² − 4ac)) / 2a works for every quadratic. It is printed on the NSC information sheet, but know it well enough to substitute quickly and to recognise the discriminant inside it.
Example (NSC Nov 2024 P1 Q1.2): solve 2x² − 3x − 7 = 0 to two decimal places.
a = 2, b = −3, c = −7. Substitute: x = [3 ± √(9 + 56)] / 4 = [3 ± √65] / 4. So x ≈ 2.77 or x ≈ −1.27. Always keep full precision until the last step, then round.
Method 3 — Completing the square
Less common in the exam as a direct instruction, but the marker can ask for it explicitly — and it's how you derive the vertex form of a parabola. To solve x² + 6x − 7 = 0: move the constant (x² + 6x = 7), add (b/2)² = 9 to both sides ((x + 3)² = 16), then square-root (x + 3 = ±4), giving x = 1 or x = −7.
Method 4 — The discriminant (nature of roots)
When a Paper 1 question asks 'for which value(s) of k...', it almost always wants the discriminant Δ = b² − 4ac.
- Δ > 0 → two distinct real roots
- Δ = 0 → equal (repeated) real roots
- Δ < 0 → non-real roots
- Δ ≥ 0 and a perfect square → rational roots
The 5 most common quadratic mistakes
- Dividing both sides by x — you lose the root x = 0.
- Forgetting the ± when taking square roots.
- Rounding the formula's intermediate values too early.
- Writing (x − 7)(x + 2) but then reading off x = 7 and x = 2 (wrong sign).
- In inequalities, forgetting to flip the sign when multiplying by a negative.
Practise on a real past paper question
Try NSC November 2024 Paper 1 Question 1 — every sub-question is a quadratic. The full memo with step-by-step working is free in our Past Papers hub. When a question stops you, snap it into Snap&Learn — you'll get a CAPS-aligned step-by-step explanation in seconds, and your first three AI solutions are free.