Graphing Quadratic Inequalities Calculator – Guide & Formulas
Graph quadratic inequalities and visualize solution sets on a number line. Enter coefficients for ax² + bx + c > 0 or < 0 and see interval solutions instantly.
Graph any quadratic inequality instantly with our free calculator. Enter a, b, and c to see the parabola, shading regions, and solution intervals on a number line.
Key Takeaway
Use the free Graphing Quadratic Inequalities Calculator to graph quadratic inequalities and visualize solution sets on a number line. enter coefficients for ax² + bx + c > 0 or < 0 and see interval solutions instantly. Get instant results with step-by-step explanations.
How to Use the Graphing Quadratic Inequalities Calculator
- Enter coefficients a, b, and c for the quadratic expression ax² + bx + c.
- Select the inequality type: >, ≥, <, or ≤.
- Review the roots (where the parabola crosses the x-axis).
- Check the solution intervals displayed on the number line.
The Formula
Variable Definitions
- a: Leading coefficient of the quadratic (x² term)
- b: Coefficient of the linear term (x)
- c: Constant term
- Δ: Discriminant: b² - 4ac. Determines root count.
- R: The relational operator: >, ≥, <, or ≤
Solving x² - 4x + 3 > 0
Find where the quadratic is positive.
- Step 1: Identify a = 1, b = -4, c = 3.
- Step 2: Find roots: x = (4 ± √(16 - 12)) / 2 = (4 ± 2) / 2. Roots: x = 1 and x = 3.
- Step 3: Since a > 0, parabola opens upward. Test intervals: (-∞,1), (1,3), (3,∞).
- Step 4: At x = 0: 0 - 0 + 3 = 3 > 0 ✓. At x = 2: 4 - 8 + 3 = -1 < 0 ✗. At x = 4: 16 - 16 + 3 = 3 > 0 ✓.
- Step 5: Solution: x < 1 or x 3 (i.e., (-∞, 1) ∪ (3, ∞)).
Frequently Asked Questions
What is a quadratic inequality?
A quadratic inequality is an inequality involving a quadratic expression, such as ax² + bx + c > 0. Instead of finding a single value, you find ranges of x that satisfy the inequality.
How do I graph a quadratic inequality?
First graph the parabola y = ax² + bx + c. Then shade above the x-axis for > or ≥ (where y is positive), or below for < or ≤ (where y is negative). The x-intercepts divide the number line into test intervals.
What does the solution set look like?
The solution is typically written as union of intervals, e.g., (-∞, 1) ∪ (3, ∞). On a number line, use open circles for strict inequalities (> or <) and closed circles for non-strict (≥ or ≤).
How do I know which direction to shade?
Check the sign of a. If a > 0, the parabola opens up, so the quadratic is positive outside the roots. If a < 0, it opens down, so positive is between the roots. Always verify by testing a point.
What if there are no real roots?
If the discriminant b² - 4ac < 0, the parabola never crosses the x-axis. If a > 0 and the inequality is > 0, all real numbers are solutions. If the inequality is < 0, there is no solution.
What if the discriminant is zero?
There is exactly one repeated root. The parabola touches the x-axis at one point. For strict inequalities (> or <), that point is excluded. For non-strict (≥ or ≤), it is included.
How is this different from solving a quadratic equation?
A quadratic equation (= 0) gives exact root values. A quadratic inequality gives intervals or ranges of values that satisfy the inequality condition.
Can I use this for non-integer coefficients?
Yes. The calculator works with any real coefficients — integers, decimals, or fractions. Just enter the values for a, b, and c.