# Graphing Quadratic Inequalities Calculator

URL: https://speedycalculator.com/calculators/math/graphing-quadratic-inequalities/
Last updated: 2026-10-30

> The quadratic inequality calculator solves ax² + bx + c (, ≥) d, finds the boundary points and writes the solution set in interval notation.

## Inputs

- a
- b
- c
- Inequality sign
- d (right-hand side)

## Results

- Solution set
- Boundary points
- Shape of the parabola

## Formula

```
Solve ax² + bx + (c − d) = 0 for the boundary points; the parabola is above or below the axis between and outside them
```

## Explanation

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).

## 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.

## FAQ

**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).


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Intermediate Algebra 2e](https://openstax.org/details/books/intermediate-algebra-2e)