Graphing Quadratic Inequalities Calculator
Enter the coefficients a, b, c and choose an inequality operator to instantly solve ax² + bx + c op 0, see the parabola, and read off the solution set with interval notation.
Inequality (right side is 0)
- 1
b² (square middle coefficient)
-1² = 1 - 2
4 × a × c
4 × 1 × -6 = -24 - 3
Discriminant D = b² − 4ac
1 − -24 = 25D > 0 → two real roots; D = 0 → one double root; D < 0 → no real roots.
How does this calculator work?
Find roots with x = (−b ± √(b²−4ac)) / 2a. For a > 0 the parabola is above the x-axis outside the roots and below between them; for a < 0 it is reversed. Strict inequalities exclude the roots (open endpoints); ≥ / ≤ include them. No real roots means the entire parabola is either always above or always below the axis.
Formula
How this is calculated
A quadratic inequality asks where the parabola y = ax² + bx + c lies above (> 0), on or above (≥ 0), below (< 0), or on or below (≤ 0) the x-axis. The first step is always to find the real roots using the quadratic formula x = (−b ± √D) / 2a where D = b² − 4ac, the discriminant.
When D > 0 there are two real roots x₁ ≤ x₂. For a > 0 (upward parabola) the curve dips below the x-axis between x₁ and x₂ and rises above it outside; for a < 0 (downward parabola) this is reversed. When D = 0 the parabola touches the x-axis exactly once at the vertex, so the strict inequalities > and < have either no solution or all-reals-except-one-point solutions. When D < 0 there are no real roots: the parabola is entirely above the axis if a > 0 (so ax² + bx + c > 0 for all x) or entirely below if a < 0.
Boundary points follow the same open/closed convention as 1D inequalities: strict operators (> and <) exclude the roots, while ≥ and ≤ include them. The graph lets you verify the result visually — the shaded portion of the x-axis corresponds to the solution region.
Frequently asked questions
Identify where the curve satisfies the inequality: for ax² + bx + c > 0 look for x-values where the parabola is above the x-axis (y > 0). The roots (where the curve crosses or touches y = 0) are the boundary points. For a > 0 the solution is outside the roots; for a < 0 it is between them.
A negative discriminant means no real roots, so the parabola never crosses the x-axis. If a > 0 (opens upward) the parabola is entirely above the x-axis, so ax² + bx + c > 0 is true for all real numbers and ax² + bx + c < 0 has no solution. If a < 0 (opens downward) the reverse is true.
Subtract d from both sides to get ax² + bx + (c − d) > 0, then enter a, b, and c − d into this calculator. The structure and method are identical — you are just moving everything to one side first.
Also known as
TG we-Calculate Editorial Team. (2026). Graphing Quadratic Inequalities Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/graphing-quadratic-inequalities-calculator
TG we-Calculate Editorial Team. "Graphing Quadratic Inequalities Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/graphing-quadratic-inequalities-calculator.
TG we-Calculate Editorial Team, "Graphing Quadratic Inequalities Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/graphing-quadratic-inequalities-calculator
@misc{wecalculate_graphing_quadratic_inequalities_calculator, title = {Graphing Quadratic Inequalities Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/graphing-quadratic-inequalities-calculator}}, year = {2026}, note = {TG we-Calculate} }
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