Graphing Quadratic Inequalities
Solve and graph quadratic inequalities like ax^2+bx+c > 0 or <= 0. Find the solution interval, boundary roots, and number-line representation for any inequality sign.
About This Calculator
The Graphing Quadratic Inequalities Calculator solves inequalities of the form ax^2 + bx + c > 0, >= 0, < 0, or <= 0. It finds the boundary roots by solving the corresponding quadratic equation ax^2 + bx + c = 0 using the quadratic formula, then determines the solution intervals based on the parabola's direction (determined by the sign of coefficient a) and the chosen inequality sign. This tool is designed for algebra students from high school through college, SAT/ACT prep, and professionals who need to quickly analyze where a quadratic function satisfies a given inequality.
The solution is derived by analyzing the sign of the quadratic expression across the number line. First, the discriminant Delta = b^2 - 4ac determines whether the parabola has zero, one, or two real x-intercepts. When the parabola opens upward (a > 0), the quadratic expression is positive outside the root interval and negative between the roots. The opposite holds when a opens downward (a < 0). If the discriminant is negative (no real roots), the entire parabola is either always positive (a > 0) or always negative (a < 0), making the solution all real numbers or empty accordingly.
The output clearly describes the solution in inequality and interval notation. For strict inequalities (>, <), the boundary roots are excluded; for non-strict inequalities (>=, <=), the boundary roots are included in the solution set. The calculator also displays the boundary roots and the discriminant-based reasoning so you can verify the logic step by step.
Regional Notes: Quadratic inequalities are taught globally in secondary and post-secondary algebra curricula. This calculator uses the standard quadratic formula and interval notation (parentheses and brackets) consistent with US, UK, and Indian mathematics education standards.
Frequently Asked Questions
How do you solve a quadratic inequality?
Rewrite as ax^2 + bx + c (sign) 0, find the roots using the quadratic formula, determine the sign of a to see if the parabola opens up or down, then shade the solution region based on the inequality sign.
What are the four types of quadratic inequalities?
ax^2 + bx + c > 0 (greater than zero), ax^2 + bx + c >= 0 (greater than or equal to zero), ax^2 + bx + c < 0 (less than zero), and ax^2 + bx + c <= 0 (less than or equal to zero).
How do you graph a quadratic inequality?
Graph the parabola y = ax^2 + bx + c as a solid curve (for >= or <=) or dashed curve (for > or <). Then shade the region above the parabola (for > or >=) or below (for < or <=).
What happens when the discriminant is negative for an inequality?
If the quadratic has no real roots (Delta < 0), the parabola never crosses the x-axis. If a > 0, the quadratic is always positive, so ax^2+bx+c > 0 is all real numbers and < 0 has no solution.
How do you write the solution in interval notation?
For x < r1 or x > r2, write (-∞, r1) union (r2, ∞). For r1 < x < r2, write (r1, r2). Use brackets [ ] for inclusive inequalities (>= or <=) and parentheses ( ) for strict inequalities (> or <).
What is the test point method?
After finding the boundary roots, pick a test point from each interval created by the roots. Substitute into the inequality. If it satisfies, that interval is part of the solution set.
Can quadratic inequalities have no solution?
Yes. For example, x^2 + 1 < 0 has no real solution because x^2 + 1 is always >= 1. Similarly, -x^2 - 1 > 0 has no solution since -x^2 - 1 is always <= -1.
Is this calculator free?
Yes, all calculators on Calculy are completely free to use with no limits or registration required.