Graphing Quadratic Inequalities Calculator

A quadratic inequality is a mathematical statement comparing a quadratic expression (degree 2 polynomial) to another value using an inequality sign such as >, ≥, <, or ≤. The most common form is ax² + bx + c > 0, where a ≠ 0. Unlike equations, inequalities have solution sets — ranges of x values — rather than just discrete points.

To solve ax² + bx + c > d graphically: (1) Rewrite it as ax² + bx + (c − d) > 0. (2) Plot the parabola y = ax² + bx + (c − d). (3) Find where the parabola crosses the x-axis (the roots). (4) For > 0, the solution is where the parabola lies above the x-axis; for < 0, where it lies below. The roots define the boundary points of the interval.

On a number line, mark the two roots x₁ and x₂ as open circles (for strict inequalities > or <) or closed circles (for ≥ or ≤). Shade the region between the roots if the parabola opens upward and the inequality is < 0, or shade outside the roots if the inequality is > 0. On a coordinate plane, shade the region above or below the parabola depending on the sign.

The discriminant D = b² − 4ac determines how many real roots exist. If D > 0, there are two distinct real roots and the solution is a finite interval or two rays. If D = 0, there is exactly one real root (a repeated root) and the solution is either all real numbers (except one point) or just a single point. If D < 0, there are no real roots and the solution is either all real numbers or the empty set, depending on the direction of the inequality and the sign of a.

Rewrite x² < 1 as x² − 1 < 0. The parabola y = x² − 1 has roots at x = −1 and x = 1. Since a = 1 > 0, the parabola opens upward, so it lies below the x-axis between the roots. Therefore, the solution is −1 < x < 1.

When a < 0, the parabola opens downward. This reverses the direction of the solution set compared to a positive 'a'. For ax² + bx + c > 0 with a < 0 and two real roots x₁ < x₂, the solution is the interval between the roots (x₁, x₂), because the downward parabola is positive only between its roots.

To graph a system of two or more quadratic inequalities, graph each parabola separately and shade the region satisfying each inequality individually. The solution to the system is the intersection (overlap) of all shaded regions. Points in the overlapping area satisfy all inequalities simultaneously.

Yes. If the discriminant is negative (no real roots) and a > 0, then ax² + bx + c > 0 for all real x (solution is all reals), while ax² + bx + c < 0 has no solution. Conversely, if a < 0 and D < 0, then ax² + bx + c < 0 for all real x. If D = 0, the inequality ax² + bx + c ≥ 0 with a > 0 has exactly one point where equality holds, and the inequality is satisfied everywhere else.