Graphing Quadratic Inequalities Calculator

Understanding Quadratic Inequalities

A quadratic inequality is an inequality that involves a quadratic expression. It takes the form ax² + bx + c > 0, ax² + bx + c ≥ 0, ax² + bx + c < 0, or ax² + bx + c ≤ 0. Solving these inequalities means finding all values of x that make the inequality true.

Solution Method

Step 1: Find Roots

Use the quadratic formula to find where the parabola crosses the x-axis.

x = (-b +/- sqrt(b^2-4ac)) / 2a

Step 2: Determine Intervals

The roots divide the number line into intervals to test.

(-inf, r1), (r1, r2), (r2, +inf)

Step 3: Test Each Interval

Pick a test point in each interval and check the sign of the quadratic.

f(test) > 0 or f(test) < 0?

Step 4: Write Solution

Combine intervals where the inequality holds true.

Union of valid intervals

Key Concepts

The Discriminant

The discriminant D = b² - 4ac determines the nature of the roots. If D > 0, there are two distinct real roots. If D = 0, there is one repeated root. If D < 0, there are no real roots and the parabola does not cross the x-axis.

Parabola Direction

When a > 0, the parabola opens upward (U-shape), meaning the quadratic is negative between the roots and positive outside. When a < 0, the parabola opens downward, and the signs are reversed.

Special Cases

No real roots (D < 0): The solution is either all real numbers or empty, depending on the sign of a and the inequality type.
One repeated root (D = 0): The parabola only touches the x-axis at one point.
When a = 0: The inequality reduces to a linear inequality bx + c > 0.

Applications

Projectile motion: determining when an object is above a certain height
Optimization problems in economics (profit margins, cost analysis)
Engineering constraints for structural design
Statistics and probability distributions
Signal processing and control systems