Quadratic Inequality Calculator
Solve inequalities of the form ax² + bx + c > 0 (or <, ≥, ≤). Enter the coefficients and choose the inequality sign to find the solution interval.
Understanding Quadratic Inequalities
A quadratic inequality asks where a parabola sits above or below the x-axis. The quadratic ax² + bx + c graphs as a U-shaped curve (if a > 0) or an upside-down U (if a < 0). The inequality symbol determines which part of the x-axis you care about.
To solve, first find where the parabola crosses the x-axis by solving ax² + bx + c = 0. These crossing points are the critical values that divide the number line into regions. The parabola is positive in some regions and negative in others.
Once you have the roots, pick a test point in each region and plug it into the inequality. Whichever regions make the inequality true form your solution set. For example, if the roots are x = 1 and x = 4, test x = 0, x = 2, and x = 5 to see which intervals satisfy the original inequality.
The Role of the Leading Coefficient
The sign of the coefficient a determines whether the parabola opens upward or downward. If a > 0, the parabola is U-shaped, starting high on both ends and dipping down in the middle. If a < 0, it's flipped, forming an upside-down U that peaks in the middle and falls off on the sides.
This shape dictates where the quadratic is positive or negative. For a > 0, the quadratic is positive outside the roots and negative between them. For a < 0, it's the reverse: negative outside, positive between. Knowing this pattern lets you write the solution interval immediately after finding the roots.
When the discriminant is negative (no real roots), the parabola never crosses the x-axis. If a > 0 and you're solving ax² + bx + c > 0, the answer is all real numbers because the parabola is entirely above the axis. If you're solving ax² + bx + c < 0 in the same case, there's no solution.
Practical Applications and Examples
Quadratic inequalities model scenarios where a relationship involves a squared term and you need a range, not a single answer. Physics problems often ask when a projectile is above a certain height. Revenue models might ask when profit (a quadratic function of price) exceeds a threshold.
Consider a simple example: x² - 5x + 6 < 0. Factor it as (x - 2)(x - 3) < 0. The roots are x = 2 and x = 3. Since the leading coefficient is positive, the parabola is positive outside these roots and negative between them. The solution to the inequality is 2 < x < 3.
For x² - 5x + 6 ≥ 0 with the same quadratic, you want where it's positive or zero. That's x ≤ 2 or x ≥ 3. Notice how changing the inequality sign flips the solution from the inside region to the outside regions. This calculator handles all such cases, giving you the interval notation automatically.
Frequently Asked Questions
What is a quadratic inequality?
A quadratic inequality has the form ax² + bx + c > 0 (or <, ≥, ≤). You're finding the range of x values that make the inequality true, not just a single number.
How do you solve a quadratic inequality?
First find the roots by solving ax² + bx + c = 0. These roots divide the number line into intervals. Test a value in each interval to see where the inequality holds.
What if the discriminant is negative?
When b² - 4ac < 0, the quadratic has no real roots. If a > 0, the parabola opens upward and is always positive or always negative depending on the inequality.
What do the critical points represent?
Critical points are the roots of the equation ax² + bx + c = 0. They mark the boundaries where the quadratic changes from positive to negative or vice versa.
How is interval notation written?
Use parentheses ( ) for strict inequalities (< or >) and brackets [ ] for inclusive inequalities (≤ or ≥). For example, (2, 5) means 2 < x < 5, while [2, 5] means 2 ≤ x ≤ 5.