Quadratic Inequality Calculator (solve & describe the graph of ax² + bx + c > 0)

Understanding the quadratic inequality calculator — concept and principle

This calculator solves a quadratic inequality ax² + bx + c > 0 (or <, ≥, ≤) and describes its graph. It gives the solution in interval and inequality notation, the roots, the discriminant, the vertex, which way the parabola opens, and how to draw the boundary and shading for the two-variable version. Use it for algebra homework and graphing practice.

How it is calculated

1. Solve ax² + bx + c = 0; the roots split the number line into intervals.
2. The sign of ax² + bx + c is constant on each interval; check one test point in each.
3. With b² − 4ac > 0: if a > 0 the expression is positive outside the roots and negative between them; a < 0 reverses this.
4. With ≥ or ≤, include the roots (square brackets); with > or <, exclude them.

• If b² − 4ac < 0 the parabola never meets the x-axis, so the inequality holds for every x or for none.
• For y > ax² + bx + c, draw the parabola dashed and shade above; solid for ≥ or ≤.

The principle

A quadratic expression can change sign only where it equals zero, at its roots. Between and outside the roots its sign is fixed, so solving the inequality means asking where the parabola lies above or below the x-axis. An upward parabola is below the axis only between its roots, which explains the rule in step 3.

Worked example

Default inputs: x² − 2x − 8 > 0.

1. Discriminant = (−2)² − 4 × 1 × (−8) = 36; roots x = −2; x = 4.
2. Vertex at x = 1: y = 1 − 2 − 8 = −9, so (1, −9); the parabola opens upward.
3. An upward parabola is positive outside its roots: x < −2; x > 4.
4. Interval notation: (−∞, −2) ∪ (4, ∞); the roots are not included.
5. Graph of y > x² − 2x − 8: dashed parabola, shading above the parabola.

How to read the result

Test a point to be sure: x = 0 gives −8, not > 0, so 0 is correctly excluded; x = 5 gives 7 > 0, included. The vertex is the lowest point here, at −9, which is why the middle interval is negative.

Common mistakes

• Forgetting to flip the reasoning when a < 0.
• Including the roots for a strict inequality.
• Writing the union of two intervals as one interval.
• Concluding "no solution" when the discriminant is negative without checking the sign of a.

Frequently asked questions

How do you solve a quadratic inequality?

Find the roots, then decide on which intervals the parabola is above or below the x-axis.

When is the solution all real numbers?

When the parabola never crosses the x-axis and lies entirely on the required side.

Dashed or solid boundary?

Dashed for < or >, solid for ≤ or ≥.

Sources: OpenStax, College Algebra 2e (2021), §2.7 Linear Inequalities and Absolute Value Inequalities, and §5.1 Quadratic Functions