Graphing Quadratic Inequalities Calculator
Solve quadratic inequalities by graphing and find solution intervals with step-by-step explanations
Enter Quadratic Inequality
Coefficient of x²
Coefficient of x
Constant term
Right-hand side
Current Inequality:
Solution
Interval Notation
Description
Quadratic Analysis
Step-by-Step Solution
Step 1: Identify the quadratic function and line
- • Quadratic function: y = 1x² + 0x + 0
- • Horizontal line: y = 0
- • Parabola opens upward (a = 1)
Step 2: Find intersection points
Solve: 1x² + 0x + 0 = 0
Rearrange: 1x² + 0x + 0 = 0
Discriminant: Δ = b² - 4ac = 0.000
One intersection point (tangent): x = 0.000
Step 3: Determine solution region
Looking for where the parabola is above the line y = 0
The parabola touches the line at exactly one point.
Final Answer
Solution: x ∈ ℝ \ {0.000}
Interval notation: (-∞, 0.000) ∪ (0.000, ∞)
Example Problems
Example 1
x² - 4 > 0
Solution: x < -2 or x > 2
Example 2
-x² + 3x - 2 ≥ 0
Solution: 1 ≤ x ≤ 2
Example 3
x² + 2x + 3 < 2
Solution: No solution
Solution Types
Two Intervals
When parabola crosses line at two points
Single Interval
Between intersection points
All Real Numbers
When parabola is entirely above/below line
No Solution
When inequality is never satisfied
Quick Tips
If a > 0, parabola opens upward
If a < 0, parabola opens downward
Use discriminant to find number of roots
Pay attention to strict vs. non-strict inequalities
Understanding Quadratic Inequalities
What are Quadratic Inequalities?
A quadratic inequality is an expression involving a quadratic trinomial (polynomial of degree 2) compared to another expression, most often zero. The general form is ax² + bx + c > d, where > can be replaced with ≥, <, or ≤.
Solution Methods
- •Graphical method (visualizing parabola and line)
- •Algebraic method (factoring and sign analysis)
- •Test point method
Graphical Solution Steps
Step 1
Draw the parabola y = ax² + bx + c
Step 2
Draw the horizontal line y = d
Step 3
Find intersection points by solving ax² + bx + (c-d) = 0
Step 4
Identify regions where inequality is satisfied