Solve and analyze quadratic inequalities of the form ax² + bx + c 0, ax² + bx + c ≤ 0, or ax² + bx + c ≥ 0.
Quadratic Inequality Solver
Enter the coefficient of the x² term. Must not be zero.
Enter the coefficient of the x term.
Enter the constant term.
< (Less Than)
> (Greater Than)
≤ (Less Than or Equal To)
≥ (Greater Than or Equal To)
Select the type of inequality.
Results
Enter values to see results
Roots of the corresponding equation (ax² + bx + c = 0):N/A
Vertex (x, y):N/A
Parabola Opens:N/A
Solution Set:N/A
Formula Explanation:
The quadratic inequality is solved by first finding the roots of the corresponding quadratic equation ax² + bx + c = 0 using the quadratic formula: x = [-b ± sqrt(b² – 4ac)] / 2a. The discriminant (Δ = b² – 4ac) determines the nature of the roots. The vertex of the parabola y = ax² + bx + c is at x = -b / 2a. The direction the parabola opens depends on the sign of 'a'. These elements help define the intervals where the inequality holds true.
Table of Key Values
Quadratic Inequality Parameters
Parameter
Value
Description
Coefficient 'a'
N/A
Coefficient of x²
Coefficient 'b'
N/A
Coefficient of x
Constant 'c'
N/A
Constant term
Discriminant (Δ)
N/A
b² – 4ac
Roots (x₁, x₂)
N/A
Solutions to ax² + bx + c = 0
Vertex x-coordinate
N/A
-b / 2a
Vertex y-coordinate
N/A
f(-b / 2a)
Graph of the Parabola
Chart Explanation:
The chart displays the parabola y = ax² + bx + c. The x-axis represents the values of x, and the y-axis represents the value of the quadratic expression. The roots are where the parabola intersects the x-axis. The shaded region (conceptually) indicates the solution set for the inequality based on the selected type (e.g., above the x-axis for '>' or '≥', below for '<' or '≤').
Understanding Quadratic Inequalities
A quadratic inequality calculator is an invaluable tool for anyone dealing with mathematical expressions involving quadratic functions. Unlike simple equations, inequalities define a range of values rather than specific points. This makes them crucial in various fields, from optimization problems in engineering and economics to understanding the behavior of physical systems. This comprehensive guide will demystify quadratic inequalities, explain their mathematical underpinnings, and show you how to effectively use our calculator.
What is a Quadratic Inequality?
A quadratic inequality is a mathematical statement that compares a quadratic expression (an expression of the form ax² + bx + c, where 'a' is not zero) to another value using inequality symbols like , ≤, or ≥. Essentially, it asks: "For which values of x does the quadratic function ax² + bx + c produce a result that is less than, greater than, less than or equal to, or greater than or equal to a certain value (often zero)?"
Who should use it?
Students: High school and college students learning algebra and pre-calculus will find this tool essential for homework, studying, and exam preparation.
Engineers & Scientists: When modeling physical phenomena or designing systems, understanding the range of conditions under which certain behaviors occur often involves solving quadratic inequalities.
Economists & Financial Analysts: Analyzing market trends, cost functions, or profit margins can lead to quadratic inequalities that need solving to determine optimal operating ranges.
Anyone learning mathematics: A solid grasp of inequalities is fundamental to advanced mathematical concepts.
Common Misconceptions:
Mistaking inequalities for equations: An equation (e.g., ax² + bx + c = 0) has specific solutions (roots), while an inequality (e.g., ax² + bx + c < 0) typically has a range or set of solutions.
Ignoring the sign of 'a': The sign of the leading coefficient 'a' dictates whether the parabola opens upwards or downwards, which fundamentally changes the solution set for inequalities.
Forgetting the 'equal to' part: For ≤ and ≥ inequalities, the roots themselves are part of the solution set, which must be reflected in the notation (e.g., using brackets or closed intervals).
Quadratic Inequality Formula and Mathematical Explanation
Solving a quadratic inequality like ax² + bx + c < 0 involves several key steps, primarily centered around the roots of the corresponding quadratic equation ax² + bx + c = 0 and the shape of the parabola y = ax² + bx + c.
Step 1: Rewrite as an Equation
First, consider the related equation: ax² + bx + c = 0.
Step 2: Find the Roots
Use the quadratic formula to find the roots (also called critical points or boundary points) of this equation:
x = [-b ± √(b² – 4ac)] / 2a
The term inside the square root, Δ = b² – 4ac, is called the discriminant.
If Δ > 0, there are two distinct real roots.
If Δ = 0, there is exactly one real root (a repeated root).
If Δ < 0, there are no real roots (the parabola does not cross the x-axis).
Step 3: Analyze the Parabola's Shape
The sign of the coefficient 'a' determines the direction the parabola opens:
If a > 0, the parabola opens upwards (U-shaped).
If a < 0, the parabola opens downwards (∩-shaped).
The vertex of the parabola occurs at x = -b / 2a.
Step 4: Determine the Solution Intervals
The roots divide the number line into intervals. We test a value from each interval (or consider the parabola's shape relative to the x-axis) to see if it satisfies the original inequality.
If a > 0 (opens up):
For ax² + bx + c > 0 or ≥ 0: The solution is outside the roots (x x₂).
For ax² + bx + c < 0 or ≤ 0: The solution is between the roots (x₁ < x < x₂).
If a < 0 (opens down):
For ax² + bx + c > 0 or ≥ 0: The solution is between the roots (x₁ < x < x₂).
For ax² + bx + c < 0 or ≤ 0: The solution is outside the roots (x x₂).
Important Note: For inequalities with '≤' or '≥', the roots themselves are included in the solution set. For ", the roots are excluded.
Variables Table:
Quadratic Inequality Variables
Variable
Meaning
Unit
Typical Range
a, b, c
Coefficients of the quadratic expression ax² + bx + c
Dimensionless
Real numbers (a ≠ 0)
x
The independent variable
Dimensionless
Real numbers
Δ (Discriminant)
b² – 4ac
Dimensionless
Real numbers
Roots (x₁, x₂)
Solutions to ax² + bx + c = 0
Dimensionless
Real or Complex numbers
Vertex
The minimum or maximum point of the parabola
(x, y) coordinates
Real numbers
Practical Examples (Real-World Use Cases)
Let's explore some scenarios where quadratic inequalities are applied.
Example 1: Projectile Motion
A ball is thrown upwards with an initial velocity of 20 m/s from a height of 5 meters. The height (h) in meters after t seconds is given by the formula h(t) = -4.9t² + 20t + 5. We want to find the time interval during which the ball is at least 25 meters high.
Inequality: -4.9t² + 20t + 5 ≥ 25
Rewritten: -4.9t² + 20t – 20 ≥ 0
Using the calculator:
a = -4.9
b = 20
c = -20
Inequality Type: ≥ (Greater Than or Equal To)
Calculator Output (simulated):
Roots: Approximately t ≈ 1.28 seconds and t ≈ 2.80 seconds.
Vertex: (2.04, 15.41)
Parabola Opens: Downwards (since a < 0)
Solution Set: [1.28, 2.80] (approximately)
Interpretation: The ball is at or above 25 meters between approximately 1.28 seconds and 2.80 seconds after being thrown.
Example 2: Business Profit Maximization
A company manufactures widgets. The profit P (in thousands of dollars) is related to the number of units sold (x, in thousands) by the function P(x) = -x² + 10x – 9. The company wants to know how many units they need to sell to make a profit of at least $12,000 (P ≥ 12).
Inequality: -x² + 10x – 9 ≥ 12
Rewritten: -x² + 10x – 21 ≥ 0
Using the calculator:
a = -1
b = 10
c = -21
Inequality Type: ≥ (Greater Than or Equal To)
Calculator Output (simulated):
Roots: x = 3 and x = 7
Vertex: (5, 4)
Parabola Opens: Downwards (since a < 0)
Solution Set: [3, 7]
Interpretation: To achieve a profit of $12,000 or more, the company must sell between 3,000 and 7,000 widgets (inclusive).
How to Use This Quadratic Inequality Calculator
Our quadratic inequality calculator is designed for ease of use. Follow these simple steps:
Input Coefficients: Enter the values for the coefficients 'a', 'b', and 'c' of your quadratic expression ax² + bx + c into the respective input fields. Remember that 'a' cannot be zero for it to be a quadratic inequality.
Select Inequality Type: Choose the correct inequality symbol (, ≤, or ≥) from the dropdown menu that matches your problem.
Calculate: Click the "Calculate" button.
How to Read Results:
Primary Result: This highlights the solution set in interval notation, indicating the range(s) of 'x' that satisfy the inequality.
Roots: These are the x-values where the corresponding equation ax² + bx + c = 0 holds true. They are the boundary points for your solution intervals.
Vertex: The coordinates (x, y) of the parabola's vertex, indicating its minimum or maximum point.
Parabola Opens: Indicates whether the parabola opens upwards (a > 0) or downwards (a < 0).
Table of Key Values: Provides a detailed breakdown of the parameters used and calculated, including the discriminant and vertex coordinates.
Graph: Visualizes the parabola and helps understand where the inequality is satisfied relative to the x-axis.
Decision-Making Guidance: Use the solution set to make informed decisions. For example, if you're determining when a system is stable (e.g., a value is below a threshold), you'll look for intervals where the quadratic expression is less than that threshold.
Key Factors That Affect Quadratic Inequality Results
Several factors influence the solution set of a quadratic inequality:
Coefficient 'a': This is perhaps the most critical factor. A positive 'a' means the parabola opens upwards, while a negative 'a' means it opens downwards. This directly impacts whether the inequality is satisfied "inside" or "outside" the roots.
Coefficient 'b': Affects the position of the axis of symmetry (x = -b/2a) and thus the vertex's x-coordinate. It shifts the parabola horizontally.
Constant 'c': This determines the y-intercept of the parabola (where x=0). It shifts the parabola vertically. A larger 'c' shifts it up, potentially changing whether the parabola intersects the x-axis.
The Inequality Symbol: Whether you are looking for values greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) significantly alters the solution set. The inclusion or exclusion of the roots depends on whether equality is permitted.
The Discriminant (Δ = b² – 4ac): This value dictates the number of real roots. If Δ 0, the entire function is always positive, and for a < 0, it's always negative. This simplifies the inequality solution considerably.
Real-World Constraints: In practical applications (like time or quantity), variables often have inherent constraints (e.g., time cannot be negative). These must be considered alongside the mathematical solution. For instance, a solution like [-5, 2] for time might be restricted to [0, 2].
Frequently Asked Questions (FAQ)
Q1: What happens if the coefficient 'a' is zero?
If 'a' is zero, the expression is no longer quadratic but linear (bx + c). The inequality becomes a linear inequality, which is solved differently (e.g., bx ≥ -c).
Q2: Can a quadratic inequality have no solution?
Yes. For example, if you have x² + 1 < 0, since x² is always non-negative, x² + 1 is always positive. Thus, there are no real values of x for which this inequality is true.
Q3: Can a quadratic inequality have infinitely many solutions?
Yes. If you have x² + 1 > 0, since x² + 1 is always positive for all real x, the solution set is all real numbers (-∞, ∞).
Q4: How do I interpret the graph for inequalities?
For ax² + bx + c > 0 (with a > 0), you look for the x-values where the parabola is *above* the x-axis. For ax² + bx + c 0), you look for where it's *below* the x-axis. The logic reverses if a < 0.
Q5: What is the difference between '<' and '≤' in the solution?
The '<' symbol means the boundary points (roots) are *not* included in the solution set. The '≤' symbol means the boundary points *are* included. This is often represented using parentheses () for exclusion and brackets [] for inclusion in interval notation.
Q6: Does the calculator handle complex roots?
This calculator focuses on real number solutions for inequalities. If the discriminant (b² – 4ac) is negative, it indicates no real roots, meaning the parabola does not cross the x-axis. The calculator will indicate this and determine the solution based on the parabola's position relative to the x-axis.
Q7: How accurate are the results?
The calculator uses standard mathematical formulas and floating-point arithmetic. Results are generally accurate to several decimal places. For critical applications, always double-check with manual calculations or more specialized software.
Q8: Can I use this for inequalities like ax² + bx + c < d where d is not zero?
Yes. Simply rearrange the inequality to the standard form ax² + bx + c' < 0 by subtracting 'd' from both sides (c' = c – d), and then use the calculator with the new constant term.