Quadratic Inequality Calculator

THE SOLUTION

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Type the three coefficients, then press =.

The squares completed, and the curve drawn

Your quadratic inequality

Type it however it is printed: x² − 5x + 6 < 0, x² > 9, or x² + 6 < 5x with terms on both sides. Nothing has to be moved across first and nothing has to be factorised. Write the square as x^2.

Any arrangement — nothing needs moving first.

Tap a box, then use the keys

Get everything to one side, find where the quadratic is zero, and those roots cut the number line into pieces. On each piece the sign never changes, so testing one point settles the whole piece. Which pieces you keep depends on the symbol and on which way the parabola opens — and when the roots are irrational, they are the ends of your answer, so rounding them moves the answer itself.

Examples — tap one and it solves

The working, step by step

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This quadratic inequality calculator takes the inequality in any arrangement, keeps the roots exact, and shows the parabola that decides the answer.

Type x² − 5x + 6 < 0 and you get 2 < x < 3. Change the symbol and you get two pieces instead: x < 2 or x > 3.

What the Quadratic Inequality Calculator Is Doing

Move everything to one side and the question stops being about two expressions. It becomes a question about the sign of one: where is this quadratic below the axis, and where is it above?

The roots are the only places that sign can change, because a parabola cannot get from below the axis to above it without passing through. So they cut the line into stretches, and inside a stretch the sign is fixed.

Finding those roots is the ordinary job of the quadratic formula; what is new here is what you do with them afterwards.

Which Way It Opens Decides Everything

An upward parabola dips below the axis between its roots and sits above it outside them. A downward one does the exact opposite.

So x² − 4 > 0 gives the two outside pieces, while −x² + 4 > 0 gives the middle. Reading the answer off the symbol without checking the sign of a is one of the two mistakes this topic is known for.

Keep the Roots Exact

Here the roots are not a step on the way to the answer — they are its endpoints.

x² − 2x − 1 > 0 has roots 1 − √2 and 1 + √2. Round the first to −0.414 and every number between −0.4142 and −0.414 changes sides. A rounded root does not make the answer tidier; it makes it a different answer.

When the Parabola Just Touches

If the discriminant is zero the curve touches the axis and turns back without crossing. That one situation gives four different answers:

< 0 → no solution — it never gets below
≤ 0 → exactly one point, the touching point itself
> 0 → every number except that point — two pieces
≥ 0 → every number

So x² − 6x + 9 ≤ 0 has the single answer x = 3. Written properly that is {3}, a set with one member, not an interval — and it is a real answer rather than a sign that something went wrong.

When It Never Crosses at All

A negative discriminant means no real roots, so there is nothing to cut the line with and no chart to build.

The sign is the same everywhere, and which sign it is depends only on which way the parabola opens. x² + 1 > 0 is true for every number; x² + 1 < 0 is true for none. Trying to factorise either one is time spent on a question the discriminant had already answered.

Open End or Closed End

The roots themselves are in when the symbol carries an equals, and out when it does not.

That is the whole rule, and it is worth applying separately from everything else: x² − 5x + 6 < 0 gives (2, 3), while the same inequality with ≤ gives [2, 3]. A two-piece answer is joined with the union sign: (−∞, 2) ∪ (3, ∞) — the same shape an or gives in a compound inequality.

Testing a Point Is Not Optional

The sign chart can be built from the shape, but the safest route is to take one number from each stretch and put it into the quadratic.

Zero is usually the easiest choice, unless it happens to be a root. Doing this catches a sign of a read backwards and a root written down wrongly, which is more than checking your own factorisation will do — and this page runs that test at every endpoint before showing you anything.

Using the Quadratic Inequality Calculator

Type it as it stands, with terms on either side and nothing moved across first. You get the collected form, the discriminant and what it means, the roots exactly, the sign in each stretch, the answer in words and in interval notation, and a drawing of the parabola with the solution marked underneath it.

Quadratic Inequality Calculator FAQ

How do you solve a quadratic inequality?

Move everything to one side, find where the quadratic is zero, and let those roots cut the number line into pieces. The sign cannot change inside a piece, so testing one point settles the whole stretch.

When is the answer between the roots and when is it outside them?

An upward parabola is below the axis between its roots and above it outside them. A downward one does the opposite, so the sign of a decides it, not the symbol alone.

What if there are no real roots?

Then the parabola never crosses the axis and the sign is the same everywhere. The answer is either every number or no number, decided by which way it opens.

What happens when the discriminant is zero?

The parabola touches the axis and turns back. That gives four different answers depending on the symbol, and one of them is a single point.

Should I round the roots?

No. Here the roots are the endpoints of the answer, so a rounded root changes which numbers fall inside the solution rather than just how it is written.

Collect, find the roots, check which way it opens — and leave the roots exact.