Graphing Quadratic Inequalities Calculator

About the Graphing Quadratic Inequalities Calculator

Quadratic inequalities extend the concept of quadratic equations by replacing the equals sign with an inequality symbol (>, <, ≥, or ≤). Instead of finding exact points where a parabola crosses the x-axis, you determine entire intervals where the parabola lies above or below the axis. This is a foundational skill in algebra that appears frequently in optimization, calculus, and real-world modeling.

To solve a quadratic inequality such as ax² + bx + c > 0, you first find the roots of the corresponding equation ax² + bx + c = 0 using the quadratic formula. These roots divide the number line into intervals. You then test a point within each interval to determine the sign of the expression. The solution is the union of all intervals where the inequality is satisfied.

The discriminant b² − 4ac determines how many real roots exist. If the discriminant is positive, the parabola crosses the x-axis at two points, creating three test regions. If zero, there is one repeated root and two regions. If negative, the parabola never crosses the axis, so the inequality is either always true or always false depending on the direction the parabola opens.

This calculator automates the entire process: compute the discriminant, find roots, identify the vertex, test each region, and display the solution in interval notation with a visual number line and sign chart. Use the eight presets to explore common inequality types.

When This Page Helps

Graphing Quadratic Inequalities Calculator helps you solve graphing quadratic inequalities problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Coefficient a, Coefficient b, Coefficient c once and immediately inspect Discriminant (b²−4ac), Roots, Vertex to validate your work.

How to Use the Inputs

1. Enter Coefficient a and Coefficient b in the input fields.
2. Select the mode, method, or precision options that match your graphing quadratic inequalities problem.
3. Read Discriminant (b²−4ac) first, then use Roots to confirm your setup is correct.
4. Try a preset such as "x²−4 > 0" to test a known case quickly.

Example Calculation

Tips & Best Practices

• If a > 0 the parabola opens upward; ">" solutions are the two outer intervals, "<" solutions are the inner interval.
• If a < 0 the parabola opens downward; the logic reverses.
• A negative discriminant means no real roots — the answer is either all reals or the empty set.
• Use test points from each region to verify the solution manually.
• Pay attention to whether the inequality is strict (<, >) or inclusive (≤, ≥) — it changes open vs. closed brackets.

How This Graphing Quadratic Inequalities Calculator Works

This calculator takes Coefficient a, Coefficient b, Coefficient c, Test point x and applies the relevant graphing quadratic inequalities relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.

Interpreting Results

Start with the primary output, then use Discriminant (b²−4ac), Roots, Vertex, Solution Interval to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.

Study Strategy

Frequently Asked Questions

• What is a quadratic inequality?

  A quadratic inequality is a second-degree polynomial expression compared to zero using an inequality sign, for example x² − 4 > 0. The solution is a set of x-values (often one or two intervals) that make the inequality true.

• How do you solve a quadratic inequality?

  Find the roots of the quadratic equation, determine the intervals they create on the number line, test a point from each interval, and select the intervals where the inequality holds.

• What does the discriminant tell me?

  The discriminant Δ = b²−4ac tells you how many real roots exist: Δ > 0 means two distinct roots, Δ = 0 means one repeated root, and Δ < 0 means no real roots.

• What is the difference between strict and non-strict inequalities?

  Strict inequalities (> or <) exclude the roots from the solution and use open parentheses in interval notation. Non-strict (≥ or ≤) include the roots and use closed brackets.

• Does the direction of the parabola matter?

  Yes. If a > 0 the parabola opens upward, and the expression is negative between the roots. If a < 0 it opens downward, and the expression is positive between the roots.

• Can this calculator handle inequalities with no real roots?

  Yes. When the discriminant is negative, the calculator determines whether the entire parabola is above or below the x-axis based on the sign of a, and reports the solution as all real numbers or the empty set.