Find the roots, then test one value in each region — a quadratic cannot change sign anywhere else. x² − x − 6 < 0 has roots at −2 and 3, and the quadratic is negative between them: (−2, 3).
Two questions share this name
“Graphing a quadratic inequality” means two different things, and running them together is the main source of confusion in the topic:
- One variable. x² − x − 6 < 0. The answer is a set of x values, drawn on a NUMBER LINE, and written in interval notation: (−2, 3).
- Two variables. y > x² − 4x + 3. The answer is a REGION of the plane, drawn as shading on one side of a curve. There is no interval notation for it.
The tool detects which one you have typed — a y on either side means the region version — and says which it is answering. They share a name and almost nothing else: different answer type, different drawing, different method.
The sign chart beats four memorised rules
Textbooks often present four cases: a > 0 with <, a > 0 with >, a < 0 with <, a < 0 with >. That works and it does not scale — it teaches nothing that survives contact with a cubic.
The sign chart does. The roots cut the line into regions; inside a region the quadratic cannot change sign, because to change sign a continuous function has to pass through zero and the roots are the only zeros. So one test value settles a whole region.
x² − x − 6 < 0 has roots at −2 and 3, giving three regions with signs +, −, + — and the middle one is what < 0 asks for. Reverse the inequality and x² − x − 6 > 0 keeps the other two: (−∞, −2) ∪ (3, ∞). Same chart, different selection.
A negative leading coefficient flips everything
−x² + 4 > 0 is positive BETWEEN its roots, not outside them: (−2, 2). The parabola opens downward, so it is above the axis in the middle and below at both ends.
This is where the four-case rule earns its reputation. “Greater than means outside the roots” is true when a is positive and false when it is negative, and remembering which way round is exactly the kind of thing that goes wrong under exam conditions.
The sign chart gets it right without any special handling, because the test value carries the sign of a with it. That is the argument for the method: it has one rule instead of four, and the one rule does not depend on remembering a case split.
No real roots still has an answer
x² + 1 > 0 has discriminant −4 — negative, so the parabola never meets the axis. That does not mean there is no answer; it means the sign is the same everywhere.
The discriminant is −4, so the parabola never meets the x-axis and its sign is the same everywhere. It opens upward, so the whole curve sits above the axis — which is exactly what the inequality asks for, so every real number works.
x² + 1 < 0 asks for the other side, and gets ∅ — the empty set. Both are correct answers to well-posed questions, and both come from a single test value, because with no roots there is only one region to test.
The repeated root, where all four relations differ
x² − 6x + 9 touches the axis at 3 without crossing it, and this is the one case where the four relations give four genuinely different answers rather than answers differing by an endpoint:
- > 0 — (−∞, 3) ∪ (3, ∞): everything except that one point.
- ≥ 0 — (−∞, ∞): everything, with no exception.
- < 0 — nothing at all. A square is never negative.
- ≤ 0 — [3, 3]: that single point and nothing else.
Four relations, four answers: the whole line, the whole line minus a point, one point, and the empty set. Elsewhere the ≤/< distinction moves an endpoint; here it changes the answer completely, which is why the repeated-root case is worth practising deliberately.
Shading the region, and choosing a test point
For the two-variable form the boundary is the parabola itself, drawn solid when the relation includes it (≤ or ≥) and dashed when it does not (< or >). Then one side gets shaded.
The boundary is the parabola y = x² − 4x + 3, drawn dashed because > excludes it. Testing (2, −2) gives −2 > −1, which is false — so shade the region above the curve. The origin is the usual test point, but it fails when the curve passes through it, so this uses a point just off the vertex instead — that one is never on the curve.
The usual advice is to test the origin, and it fails whenever the curve passes through the origin — which for a quadratic with no constant term it does. This tool tests a point just off the vertex instead, which is never on the curve for any parabola, so the choice never needs a special case.
Endpoints work the same way as on the number line: x² − 2 < 0 has irrational roots and its answer is (−√2, √2), kept as surds rather than rounded, because −√2 and √2 are exact and 1.4142135 is not.
Sources and methodology
Sign analysis of a polynomial is standard; the reference below covers the elementary theory it rests on.
Method. Every figure on this page comes from src/lib/graphing-quadratic-inequalities.ts, which reuses the exact interval-set representation from src/lib/interval-notation.ts — so the answers here and on that page are the same objects, and unions merge identically. Rational endpoints stay fractions; irrational ones carry an exact binary-fraction value for ordering and print through their surd form. That engine is verified on every change against 49 hand-written assertions, including that the selected regions always match the sign chart, and that an endpoint is included exactly when the relation is non-strict. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The answers this page produces are written in interval notation, and the Interval Notation Calculator converts them to the other three forms. For the solving step behind the roots, Which Method Should You Use to Solve a Quadratic? covers when factoring beats the formula.