Did you know? You can sometimes tell that a parabola never crosses the x-axis without plotting a single point. One small calculation, the discriminant, gives you a powerful clue. But it does not tell the whole story until you also check which way the parabola opens.
That is the distinction I want JC2 students to notice in H1 Mathematics 8865: the discriminant describes real roots; the leading coefficient helps you decide the sign of the entire quadratic.
The discriminant, in everyday language
For f(x) = ax² + bx + c with a ≠ 0, the discriminant is Δ = b² − 4ac. Think of it as a quick check on how many real solutions the equation f(x) = 0 can have.
- Δ > 0: two distinct real roots; the parabola crosses the x-axis twice.
- Δ = 0: one repeated real root; the parabola just touches the x-axis.
- Δ < 0: no real roots; the parabola does not touch the x-axis.
Do not leap from ‘no real roots’ to ‘always positive’. A downward-opening parabola with no real roots can be always negative. The sign of a matters.
Worked question 1: is the quadratic always positive?
Consider f(x) = 2x² − 4x + 5. Here a = 2, b = −4 and c = 5.
Δ = (−4)² − 4(2)(5) = 16 − 40 = −24. There are no real roots. Because a = 2 > 0, the graph opens upwards and stays above the x-axis, so f(x) > 0 for every real x.
Want a second way to see it? Complete the square: f(x) = 2(x − 1)² + 3. The squared term cannot be negative, so the minimum value is 3 when x = 1. This checks the conclusion rather than merely repeating it.
The exact wording matters: for a quadratic to be strictly positive for all real x, a > 0 and Δ < 0. If the question allows zero as well, ‘non-negative for all real x’ permits Δ = 0.
Worked question 2: where is the expression negative?
Now solve x² − 5x + 6 ≤ 0. Factorise: (x − 2)(x − 3) ≤ 0. The roots are 2 and 3, and the leading coefficient is positive.
The parabola is below or on the x-axis between the roots. Thus the answer is 2 ≤ x ≤ 3. The equal signs mean we include x = 2 and x = 3, where the expression is exactly zero.
A common wrong turn is answering ‘x = 2 or x = 3’. Those are the solutions of the equation, not the full set of values satisfying the inequality. An inequality asks for an interval.
A 30-second sign checklist
- Write the quadratic with all terms on one side.
- Identify a, b and c carefully, including negative signs.
- Use Δ when the question asks about roots or an always-positive/always-negative condition.
- Use a sign sketch, number line or graph to identify intervals for inequalities.
- Check strict (>, <) against inclusive (≥, ≤) symbols.
- Substitute a test value from your proposed interval to catch a reversed sign.
The official 2026 H1 Mathematics 8865 syllabus includes conditions for real quadratic roots, always-positive or always-negative quadratics, and solving quadratic equations and inequalities analytically. Graphical approaches to inequalities are also part of the course.
What a useful JC2 tutorial can diagnose
I would put two near-identical-looking questions side by side: ‘solve f(x) = 0’ and ‘solve f(x) ≤ 0’. Then ask the student why the first answer consists of roots while the second may be an interval. That one comparison often reveals whether the student is reasoning about a graph or only copying algebra.
For revision, practise one Δ < 0 question, one Δ = 0 question and one sign-interval question. Explain the conclusion aloud. You will soon hear exactly where your understanding slips.
Quick questions
Can I use a graphing calculator? Yes, as a check, but write the required mathematical reasoning when the question asks for an analytical solution or working.
Does Δ < 0 mean the graph is always above the axis? Only when a > 0. If a < 0, it is always below.
See the H1 Mathematics 8865 syllabus overview and JC2 paper revision map for a wider plan. For a targeted tutoring enquiry, contact eduKate Singapore with a recent quadratic question and the student’s working.