Quadratic Inequality Calculator
Solve ax² + bx + c > 0, ≥ 0, < 0 or ≤ 0: exact roots, the solution in interval and set-builder notation, and a sign chart with test points for every interval.
How it works
Solve ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a · for a > 0 the parabola is positive outside the roots and negative between them (reverse for a < 0) · strict < or > excludes the roots, ≤ or ≥ includes them
A quadratic inequality is solved by finding where the parabola y = ax² + bx + c crosses the x-axis and reading off where it lies above (> 0) or below (< 0). The roots split the number line into intervals; one test point per interval confirms the sign. If the discriminant is negative the parabola never crosses the axis, so the answer is either every real number or nothing, depending on whether it opens upward or downward.
Worked example
x² − x − 6 > 0: the roots of x² − x − 6 = (x + 2)(x − 3) are x = −2 and 3. Testing x = −3 gives 6 (+), x = 0.5 gives −6.25 (−) and x = 4 gives 6 (+). The parabola opens upward, so the solution is (−∞, −2) ∪ (3, ∞), i.e. {x | x < −2 or x > 3}.
Sources: OpenStax Intermediate Algebra 2e — 9.9 Solve Quadratic Inequalities · Paul’s Online Math Notes — Quadratic Inequalities
Frequently asked questions
How do you solve a quadratic inequality step by step?
Move everything to one side, solve the equation ax² + bx + c = 0, mark the roots on a number line, test one point in each interval and keep the intervals where the sign matches the inequality.
When do I use brackets instead of parentheses?
Use square brackets [ ] when the roots are included (≤ or ≥) and parentheses ( ) when they are not (< or >). Infinity always gets a parenthesis.
What if the discriminant is negative?
The parabola never touches the x-axis. If it opens upward, ax² + bx + c > 0 is true for all real x and < 0 has no solution; if it opens downward, the opposite.
Do I flip the inequality when a is negative?
Only if you divide or multiply both sides by a negative number. The calculator handles it for you: with a < 0 the parabola opens downward, so it is positive between the roots.
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This calculator is for educational purposes. Double-check important results before acting on them.