Graphing Quadratic Inequalities Calculator
Analyze and graph quadratic inequalities of the form ax² + bx + c op 0, with roots, vertex, solution set, and interval notation.
Enter coefficients a, b, c and choose the inequality sign to analyze the parabola and determine the solution set.
Graphing Quadratic Inequalities Calculator
Analyze and graph quadratic inequalities of the form ax² + bx + c op 0, with roots, vertex, solution set, and interval notation.
About the Quadratic Inequality Calculator
A quadratic inequality is an inequality that involves a quadratic expression — that is, a polynomial of degree 2 — compared to a value using <, ≤, >, or ≥. The most common form is ax² + bx + c > 0 or ax² + bx + c < 0, where a ≠ 0. Unlike a quadratic equation, which asks for specific values of x that make the expression equal to zero, a quadratic inequality asks for all values of x that make the expression positive, negative, non-positive, or non-negative. The answer is typically an interval or a union of intervals on the real number line.
The key to solving a quadratic inequality is understanding the parabola y = ax² + bx + c. The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0). The x-intercepts — the roots of the corresponding equation ax² + bx + c = 0 — are where the parabola crosses or touches the x-axis. The discriminant Δ = b² − 4ac tells you how many real roots exist: if Δ > 0 there are two distinct real roots, if Δ = 0 there is exactly one (a double root), and if Δ < 0 there are no real roots.
To solve ax² + bx + c > 0 when Δ > 0 and a > 0: the parabola opens upward and dips below the x-axis between its two roots. So the expression is positive outside the roots — that is, for x < r₁ or x > r₂. For the inequality < 0 with the same conditions, the solution is the interval between the roots: r₁ < x < r₂. When a < 0 the parabola opens downward and these cases are reversed.
When Δ = 0 there is a single touching point. For a > 0 the expression is ≥ 0 for all x (touching zero at the double root) and < 0 for no x. When Δ < 0 and a > 0, the parabola never crosses the x-axis and stays entirely above it, so ax² + bx + c > 0 for all real x and the inequality < 0 has no solution.
Quadratic inequalities arise in projectile motion (when is the projectile above a certain height?), optimization (for what inputs does cost exceed revenue?), signal processing (frequency bands), and engineering tolerances. The discriminant formula b² − 4ac and the quadratic formula x = (−b ± √Δ) / (2a) are the two workhorses of the analysis.
This calculator accepts the coefficients a, b, and c and the inequality sign, then computes the discriminant, finds any real roots, determines the vertex, and describes the solution set in both plain language and interval notation. The direction the parabola opens is also reported to help you visualize the graph.
Quadratic Inequality Examples
Four cases covering upward and downward parabolas, two distinct roots, and a double root.
| Inequality | Solution Set | Notes |
|---|---|---|
| x² − 4x + 3 > 0 (a=1, b=−4, c=3) | (−∞, 1) ∪ (3, ∞) | Parabola opens up, roots at x=1 and x=3. Expression is positive outside the roots. |
| −x² + 2x + 3 ≤ 0 (a=−1, b=2, c=3) | (−∞, −1] ∪ [3, ∞) | Parabola opens down, roots at x=−1 and x=3. Expression is non-positive outside the roots. |
| 2x² + 3x + 4 < 0 (a=2, b=3, c=4) | No solution | Discriminant Δ = 9 − 32 = −23 < 0 and a > 0, so the expression is always positive. |
| x² − 6x + 9 ≥ 0 (a=1, b=−6, c=9) | All real numbers | Double root at x=3 (a perfect square). The expression is zero only at x=3 and positive everywhere else. |
How to Use the Quadratic Inequality Calculator
- Enter coefficient a (the x² term), b (the x term), and c (the constant). Coefficient a must not be zero.
- Select the inequality sign from the dropdown: >, ≥, <, or ≤.
- Click 'Graph Inequality'. The calculator computes the discriminant, finds the roots (if any), locates the vertex, and determines the complete solution set.
- Read the solution set in interval notation in the result panel. A union symbol ∪ means the solution consists of two separate intervals.
- Use Reset to clear all fields and start a new problem.
Quadratic Inequality Calculator FAQ
What is a quadratic inequality?
A quadratic inequality is an inequality of the form ax² + bx + c > 0, ax² + bx + c < 0, ≥, or ≤, where a ≠ 0. Instead of finding specific values of x as in an equation, you find all values of x that satisfy the inequality — typically a range or union of ranges.
How does the sign of the leading coefficient a affect the solution?
When a > 0 the parabola opens upward, so the expression is negative between the roots and positive outside them. When a < 0 the parabola opens downward, so the expression is positive between the roots and negative outside them. Flipping the sign of a essentially flips the solution set.
What happens when the discriminant is negative?
If Δ = b² − 4ac < 0 the parabola never crosses the x-axis. When a > 0 the expression is always positive, so ax²+bx+c > 0 for all real x (solution = ℝ) and ax²+bx+c < 0 has no solution. When a < 0 the reverse is true.
What is a double root and what does it mean for the solution?
A double root occurs when Δ = 0, meaning the parabola just touches the x-axis at exactly one point. For a > 0, the expression is ≥ 0 for all x (solution is all reals for ≥) and the expression is never strictly negative (no solution for <). For the ≤ inequality with a double root at r, the solution is just the single point x = r.
How do I read the interval notation in the result?
Parentheses ( ) denote strict boundaries (not included, used for > or <), while square brackets [ ] denote inclusive boundaries (used for ≥ or ≤). The symbol ∪ means 'union' — the solution is the set of all numbers in either interval.
Can the solution be all real numbers?
Yes. If a > 0 and Δ < 0, then ax² + bx + c > 0 for all real x, so the solution to ax²+bx+c > 0 (or ≥ 0) is all of ℝ. Similarly if a < 0 and Δ < 0 then ax²+bx+c < 0 for all real x.