Solve and graph quadratic inequalities of the form ax²+bx+c > 0, < 0, ≥ 0, or ≤ 0. Find roots, vertex, solution intervals, and visualize solution regions on a number line.
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 instantly.
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.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Roots: x = (−b ± √(b²−4ac)) / 2a; Vertex: (−b/2a, f(−b/2a)); Discriminant: Δ = b²−4ac. The solution set is the union of intervals where the sign of f(x) matches the inequality.
Result: Discriminant (b²−4ac) shown by the calculator
Using the preset "x²−4 > 0", the calculator evaluates the graphing quadratic inequalities setup, applies the selected algebra rules, and reports Discriminant (b²−4ac) with supporting checks so you can verify each transformation.
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.
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.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
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.
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. Use this as a practical reminder before finalizing the result.
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. Keep this note short and outcome-focused for reuse.
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.
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.
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.