Quadratic inequality
Solve a quadratic inequality with sign regions shown on a graph.
Enter the coefficients and pick a type – the solution and graph appear instantly.
How to solve a quadratic inequality
First find the roots of the corresponding equation ax² + bx + c = 0 (using the discriminant). The roots divide the number line into segments, and in each segment the quadratic function has a constant sign. The shape of the parabola then tells us which segments satisfy the inequality.
The key is the sign of coefficient a: when a > 0, the parabola opens upward – the function is negative between the roots and positive outside. When a < 0, it's the opposite.
Step-by-step example
x² − 5x + 6 > 0: the roots are 2 and 3. The parabola opens upward, so it's positive outside the roots. The solution of > 0 is thus x ∈ (−∞, 2) ∪ (3, +∞). If we wanted < 0, the solution would be the interval between the roots, (2, 3).
When D ≤ 0
If the equation doesn't have two distinct roots (D = 0 or D < 0), the parabola either just touches the x-axis or doesn't cross it at all. Then the function doesn't change sign (except possibly at one point), and the solution is either all of ℝ, a single point, or the empty set. The calculator recognises all these cases and shows them clearly on the graph.
Tip: the union of two intervals is denoted by ∪. "Satisfies" is the highlighted part of the graph.