Quadratic Inequality Calculator
Solve ax² + bx + c < 0, >, ≤, ≥ with interval notation
Understanding Quadratic Inequalities
A quadratic inequality is an inequality involving a quadratic expression: ax² + bx + c < 0, > 0, ≤ 0, or ≥ 0. Unlike equations which have specific solutions, inequalities have solution sets—ranges of values that satisfy the inequality.
Solving Method
- Find the critical points by solving the related equation ax² + bx + c = 0
- These critical points divide the number line into intervals
- Test a point from each interval to determine if it satisfies the inequality
- Select the intervals where the inequality is true
- Write the solution in interval notation
Graphical Interpretation
The solution to a quadratic inequality corresponds to portions of the x-axis where the parabola is above or below the x-axis:
- ax² + bx + c > 0: Where the parabola is above the x-axis
- ax² + bx + c < 0: Where the parabola is below the x-axis
- Including equality (≤, ≥): Also includes the x-intercepts
Interval Notation
Solutions are written using interval notation:
- (a, b): Open interval, excludes endpoints
- [a, b]: Closed interval, includes endpoints
- (a, b]: Half-open, excludes a but includes b
- ∪: Union symbol, combines multiple intervals
- ∅: Empty set, no solution
Special Cases
No real roots (Δ < 0): The parabola doesn't cross the x-axis. If it opens upward, it's always positive. If it opens downward, it's always negative.
One root (Δ = 0): The parabola touches the x-axis at one point. It's zero at that point and positive/negative elsewhere depending on direction.
Frequently Asked Questions
Why do we test intervals?
The sign of a quadratic expression can only change at its roots (where it equals zero). Between roots, the expression maintains the same sign. Testing one point in each interval tells us the sign throughout that interval.
What's the difference between < and ≤?
< (strict inequality) excludes the boundary points (roots). ≤ (non-strict) includes them. In interval notation, this is shown by parentheses ( ) vs. brackets [ ].
How do I know which intervals to choose?
After testing, choose intervals where the test point satisfies the inequality. For < or ≤, choose negative regions. For > or ≥, choose positive regions. The parabola's direction (opening up or down) determines the sign pattern.
What if there are no real roots?
If the discriminant is negative, the parabola never crosses the x-axis. The expression is either always positive (if a > 0) or always negative (if a < 0), so the solution is either all real numbers or no solution, depending on the inequality.
Can I solve inequalities by graphing?
Yes! Graph y = ax² + bx + c and look where it's above or below the x-axis. The x-values in those regions form your solution. This visual method confirms algebraic solutions.
Why use interval notation instead of inequality notation?
Interval notation is more compact and clearer for complex solutions, especially when there are multiple disjoint intervals (using ∪). It's also the standard in higher mathematics.
What does ∪ mean in the solution?
The union symbol ∪ combines multiple separate intervals. For example, (-∞, -2) ∪ (3, +∞) means "x is less than -2 OR x is greater than 3"—two separate regions that satisfy the inequality.
How do inequalities relate to real-world problems?
Quadratic inequalities model constraints in optimization: profit must be positive, height must be above ground, concentration must be safe, costs must be below budget. They help determine feasible ranges in engineering, economics, and science.