Graphing Quadratic Inequalities Calculator
Enter your quadratic coefficients and choose the inequality direction to instantly find the solution set and interval notation.
📉 What is a Quadratic Inequality?
A quadratic inequality is a mathematical statement that compares a quadratic expression to zero using an inequality sign. The four standard forms are ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, and ax² + bx + c ≤ 0, where a ≠ 0. Unlike a quadratic equation (which finds specific points where the parabola crosses the x-axis), a quadratic inequality asks which values of x make the expression positive, negative, non-negative, or non-positive. The answer is typically a range of values, not a single number.
Quadratic inequalities appear across engineering, physics, economics, and computer science. In projectile motion, finding the time window during which an object is above a certain height reduces to solving h(t) > k, which is a quadratic inequality in t. In business, finding the range of production volumes that yield a positive profit can produce a quadratic inequality in units produced. In signal processing, analyzing the frequency ranges where a quadratic filter's gain exceeds a threshold uses the same technique.
The solution method relies on two key facts: first, the quadratic formula or discriminant tells you where (if anywhere) the expression equals zero, giving the boundary or critical points. Second, because a parabola is a smooth continuous curve, it can only change sign at these boundary points. Between any two consecutive boundary points, the expression maintains a constant sign. This means you only need to test one point in each region to know the sign of the entire region.
This calculator handles all three discriminant cases automatically. When D > 0 (two distinct real roots), it identifies the two critical points and expresses the solution as a union of intervals or a bounded interval depending on the parabola direction and inequality type. When D = 0 (one repeated root), it handles the special cases where the solution may be a single point or all reals except one point. When D < 0 (no real roots), the parabola never crosses the x-axis, so the solution is either all real numbers or the empty set.
📐 Formula
📖 How to Use This Calculator
Steps to Solve a Quadratic Inequality
💡 Example Calculations
Example 1 — Classic Upward Parabola, Positive Region
Solve x² − 5x + 6 > 0
Example 2 — Upward Parabola, Negative Region
Solve x² − 5x + 6 < 0
Example 3 — Downward Parabola
Solve −x² + 4x − 3 ≥ 0
Example 4 — No Real Roots (All Reals)
Solve x² + x + 2 > 0
❓ Frequently Asked Questions
🔗 Related Calculators
How do you solve a quadratic inequality step by step?
Step 1: Rearrange to the form ax² + bx + c > 0 (or <, >=, <=). Step 2: Solve ax² + bx + c = 0 to find the boundary roots using the quadratic formula. Step 3: Note the sign of a and the direction of the inequality to identify which region satisfies it. Step 4: Write the solution set and interval notation.
What is interval notation for a quadratic inequality solution?
Interval notation uses parentheses () for excluded endpoints and brackets [] for included endpoints. For ax² + bx + c > 0 with a > 0 and roots x1 < x2, the solution is (-∞, x1) ∪ (x2, +∞). For < 0, the solution is (x1, x2). The ∪ symbol means 'union' (combine both parts).
What happens if the discriminant is negative in a quadratic inequality?
If D < 0, the quadratic has no real roots and the parabola never crosses the x-axis. If a > 0, the entire parabola is above the x-axis: ax² + bx + c > 0 for all real x (solution = all reals), and ax² + bx + c < 0 has no solution. If a < 0, it is reversed.
What does it mean when a quadratic inequality has no solution?
No solution means there are no real values of x that satisfy the inequality. For example, x² + 1 < 0 has no solution because x² + 1 is always positive (minimum value is 1 at x = 0). The solution set is the empty set, written as ∅ or {}.
How does the parabola direction affect the solution?
When a > 0 the parabola opens upward, so the expression is positive outside the roots and negative between them. When a < 0 the parabola opens downward, so the expression is positive between the roots and negative outside. This is why the same roots produce different solution sets for different signs of a.
What is the difference between strict and non-strict quadratic inequalities?
A strict inequality (> or <) excludes the boundary points where the expression equals zero. A non-strict inequality (>= or <=) includes them. In interval notation: strict uses parentheses at boundary points, non-strict uses brackets. Example: for roots 2 and 5, strict x² - 7x + 10 > 0 gives (-∞,2) ∪ (5,+∞) while non-strict >= 0 gives (-∞,2] ∪ [5,+∞).
How do you graph a quadratic inequality?
Graph y = ax² + bx + c as a parabola. For > 0 (or >= 0), shade the region above the x-axis (y > 0). For < 0 (or <= 0), shade the region below. The x-coordinates of the shaded region on the x-axis give the solution set. For strict inequalities, use open circles at the roots; for non-strict, use filled circles.
What is the solution when the quadratic has a repeated root?
With a repeated root x₀ (D = 0), the parabola just touches the x-axis at one point. If a > 0 and the inequality is > 0, the solution is all reals except x₀. If a > 0 and the inequality is >= 0, the solution is all real numbers. If a > 0 and < 0, there is no solution. If a > 0 and <= 0, the only solution is the single point x₀.
Can a quadratic inequality have a solution of all real numbers?
Yes. If a > 0 and D < 0 (no real roots, parabola always above x-axis), then ax² + bx + c > 0 is true for all real x. Similarly, if a < 0 and D < 0, then ax² + bx + c < 0 for all real x. These are the cases where the entire number line is the solution.
How do you write the solution set of x squared minus 5x plus 6 less than 0?
First solve x² - 5x + 6 = 0: x = 2 or x = 3. Since a = 1 > 0, the parabola opens upward, so x² - 5x + 6 < 0 between the roots. Solution set: 2 < x < 3. Interval notation: (2, 3). Test point x = 2.5: 6.25 - 12.5 + 6 = -0.25 < 0, confirmed.
What is the quadratic formula used in this calculator?
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) finds the roots of ax² + bx + c = 0. The discriminant D = b² - 4ac determines the nature of roots: D > 0 gives two distinct real roots, D = 0 gives one repeated root, D < 0 gives no real roots. The roots serve as the boundary points of the inequality solution.
How do you solve a quadratic inequality with no middle term?
For ax² + c > 0 (b = 0), the roots (if real) are x = ±√(-c/a). Example: x² - 9 > 0 has roots x = ±3 and since a = 1 > 0, the solution is x < -3 or x > 3, i.e. (-∞,-3) ∪ (3,+∞). Another example: x² + 4 > 0 has D = -16 < 0, so with a > 0 the expression is always positive: solution is all reals.