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A quadratic inequality's solution depends on where its parabola crosses the x-axis and which way it opens. This tool finds the roots, then works out exactly which region — between the roots, outside them, or all/no real numbers — satisfies the inequality.
How it works
The roots are found via the quadratic formula. Since a quadratic's sign is constant between consecutive roots, the region outside the roots always matches the sign of the leading coefficient a, and the region between them is the opposite sign — from there, matching against the requested inequality gives the solution.
- Enter a (coefficient of x²).
- Enter b (coefficient of x).
- Enter c (constant).
- Enter inequality.
- Click Calculate to see your results.
Examples
A "between the roots" solution
x²-x-6 < 0 has roots -2 and 3; since a>0, the negative region is between the roots: -2 < x < 3.
An "outside the roots" solution
x²-x-6 > 0 has the same roots, but now the solution is the outside region: x < -2 or x > 3.
Who should use it
- Algebra 2 and precalculus coursework on quadratic inequalities.
- Checking sign-analysis homework problems.
Industry applications
- Mathematics education
Advantages
- Correctly handles all four inequality types plus the no-real-root and repeated-root edge cases.
- Shows both the description and interval notation.
Limitations
- Only handles quadratics — cubic or higher-degree polynomial inequalities need different techniques.
Common mistakes to avoid
- Assuming the solution is always "between the roots" — that's only true for a specific combination of the leading coefficient's sign and the inequality direction.
Best practices
- Sketch a rough parabola (opening up or down based on the sign of a) and mark the roots — the correct region becomes visually obvious.
Tips
- If a > 0 (parabola opens upward), the "less than 0" region is always between the roots; if a < 0, it's the opposite — memorizing this rule speeds up hand-solving considerably.