What does this calculator solve?

This calculator solves quadratic inequalities where ax²+bx+c is compared with zero. You enter a, b and c, choose <, ≤, > or ≥, and the calculator shows the discriminant, roots, sign behavior and solution set.

• For Δ > 0, it handles two real roots and three sign intervals.
• For Δ = 0, it handles the repeated-root boundary correctly.
• For Δ < 0, it handles the no-real-root case without assuming the solution is always empty.
• It gives the final answer in interval notation.
• It includes a schematic parabola sketch to support understanding, not a precision graphing tool.

What is a quadratic inequality?

A quadratic inequality compares a quadratic expression with zero. The standard form is ax²+bx+c [op] 0, where a cannot be zero and the operator is one of <, ≤, > or ≥.

The answer is usually not a single number. It is a set of x-values, often one interval or two intervals, where the quadratic expression has the requested sign.

How does the solution method work?

The method finds the x-axis boundaries first, then checks where the parabola is positive, negative or zero. The sign intervals are selected according to the chosen operator.

1. Compute the discriminant Δ = b² − 4ac.
2. Find and order the real roots if they exist.
3. Use the sign of a to decide whether the parabola opens upward or downward.
4. Determine the sign of ax²+bx+c on each interval.
5. Include or exclude root endpoints depending on whether the operator is strict or non-strict.

How does the discriminant affect the answer?

The discriminant tells how many real boundary points the parabola has. Δ > 0 gives two real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.

• Δ > 0: the solution usually comes from either between the roots or outside the roots.
• Δ = 0: the answer may be a single point, all real numbers except that point, all real numbers, or the empty set.
• Δ < 0: there is no real boundary; the quadratic keeps the same sign for all real x-values.

What is the difference between strict and non-strict inequalities?

Strict operators, < and >, exclude root points because the expression equals zero there. Non-strict operators, ≤ and ≥, include the appropriate root points because equality is allowed.

For x²−5x+6 < 0, the answer is (2, 3). For x²−5x+6 ≤ 0, the answer is [2, 3]. The only difference is whether the endpoints are included.

Example: two real roots

For x² − 5x + 6 ≤ 0, the coefficients are a=1, b=-5 and c=6. The discriminant is 1, so the real roots are x=2 and x=3.

1. The parabola opens upward because a is positive.
2. An upward-opening parabola is negative between the two roots and positive outside them.
3. The requested sign is ≤ 0, so the region between the roots is selected.
4. Equality is allowed, so both endpoints are included.
5. Solution: [2, 3].

Example: repeated root and single-point solution

For x² − 4x + 4 ≤ 0, the expression is (x−2)². The discriminant is 0 and the repeated root is x=2.

A square is never negative; it is only zero at x=2. Therefore ≤ 0 gives the single-point solution {2}. If the operator were < 0, the solution would be the empty set.

Example: no real roots

For x² + 1 > 0, the discriminant is -4, so there are no real roots. Since a is positive, the parabola stays above the x-axis for every real x.

The solution is all real numbers. If the inequality were x² + 1 < 0, the solution would be empty because the expression is never negative.

How to read the graph and sign chart

The schematic graph shows whether the parabola opens upward or downward, where the real roots are, and which parts of the x-axis belong to the solution.

• If a is positive, the parabola opens upward.
• If a is negative, the parabola opens downward.
• Root markers show possible interval endpoints.
• Highlighted x-axis segments show where the inequality is true.
• The graph supports the interval answer but does not replace the interval notation.

When is this calculation useful?

Quadratic inequality solving is useful whenever you need to know where a parabola is above, below, or touching the x-axis. It is common in algebra classes, function sign analysis and graph interpretation.

This calculator is especially helpful when the roots are known or computable but the correct interval is easy to choose incorrectly. It combines the root calculation, sign logic and interval notation in one place.

Common mistakes

The most common mistake is to stop after finding the roots. In a quadratic inequality, roots are boundary points; the actual answer is the set of intervals where the requested sign is true.

• Forgetting that a negative a flips the sign pattern.
• Using brackets for a strict inequality or parentheses for a non-strict endpoint.
• Assuming Δ < 0 always means no solution.
• Treating a repeated root as a normal crossing point.
• Reading the schematic graph as if it were a precise coordinate plot.

Limitations of this calculator

This calculator is limited to quadratic inequalities in the form ax²+bx+c [op] 0. If a=0, the expression is not quadratic and the calculator rejects it.

• It does not solve linear inequalities; use a number-line inequality tool for that case.
• It does not shade two-variable regions on a coordinate plane.
• It does not provide a precision graphing or zooming interface.
• It does not solve symbolic parameter inequalities.
• It expects the expression to already be arranged as ax²+bx+c compared with zero.