Quadratic inequalities describe ranges – the times when a thrown ball is above a certain height, or the prices for which profit is positive. The solution is one interval or two rays, depending on the direction of the parabola and the inequality sign.
How to use the Graphing Quadratic Inequalities Calculator
- a
- b
- c
- Inequality sign
- d (right-hand side)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The calculator moves d to the left side, finds the roots with the quadratic formula and tests which regions satisfy the inequality, using the sign of a (which way the parabola opens).
Formula
Solve ax² + bx + (c − d) = 0 for the boundary points; the parabola is above or below the axis between and outside them
Worked example
Inputs — a: 1; b: -5; c: 6; Inequality sign: < (less than); d (right-hand side): 0.
Results — Solution set = (2, 3); Boundary points = Boundary points x = 2 and x = 3; Shape of the parabola = The parabola opens upward.
Key terms
- Boundary point
- A root where the expression equals the right-hand side.
Frequently asked questions
How do I solve x² − 5x + 6 < 0?
The roots are 2 and 3 and the parabola opens upward, so it is negative between them: (2, 3).
What if there are no real roots?
The parabola is entirely above or below the axis, so the solution is every real number or no real number.
Do I include the boundary points?
Only for ≤ and ≥.
Why does the sign of a matter?
It decides whether the parabola is below the axis between the roots (a > 0) or outside them (a < 0).