Graphing Quadratic Inequalities Calculator
Graph quadratic inequalities and find solution regions
Category: Mathematics
Graphing Quadratic Inequalities Calculator Inputs
Graphing Quadratic Inequalities Calculator Formula
Equation
ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, ax² + bx + c ≤ 0
Excel Formula
=ax^2+bx+c>0,ax^2+bx+c<0,ax^2+bx+c≥0,ax^2+bx+c≤0
Variables
- Coefficient of x² (a) — Leading coefficient
- Coefficient of x (b) — Linear coefficient
- Constant term (c) — Constant term
- Inequality Type — Type of inequality
How the Graphing Quadratic Inequalities Calculator Works
Graphing quadratic inequalities involves finding the solution set of expressions in the form ax^2 + bx + c > 0, ax^2 + bx + c < 0, ax^2 + bx + c >= 0, or ax^2 + bx + c <= 0. Unlike linear inequalities which have a single boundary point, quadratic inequalities typically have two critical points (the roots) that divide the number line into regions. Understanding the parabola's shape and position relative to the x-axis is essential for determining which regions satisfy the inequality. This skill is fundamental in algebra, calculus, optimization problems, and real-world applications involving quadratic relationships.
The core relationship is ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, ax² + bx + c ≤ 0. Typical inputs include Coefficient of x² (a), Coefficient of x (b), Constant term (c), Inequality Type.
Enter your values in the graphing quadratic inequalities calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online mathematics tool is built for homework, design checks, and professional verification.
Graphing Quadratic Inequalities Calculator Theory & Explanation
Understanding Quadratic Inequalities
A quadratic inequality compares a quadratic expression to zero. The solution set consists of all x-values that make the inequality true. Unlike quadratic equations which typically have 0, 1, or 2 discrete solutions, quadratic inequalities usually have solution sets that are intervals or unions of intervals. The key to solving them is understanding where the parabola y = ax^2 + bx + c is above or below the x-axis.
\beginaligned ax^2 + bx + c &> 0 \quad \text(parabola above x-axis) \\ ax^2 + bx + c &< 0 \quad \text(parabola below x-axis) \\ ax^2 + bx + c &≥ 0 \quad \text(at or above x-axis) \\ ax^2 + bx + c &≤ 0 \quad \text(at or below x-axis) \endaligned
The Quadratic Formula and Discriminant
The roots of ax^2 + bx + c = 0 are found using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
The discriminant Δ = b^2 - 4ac determines the number of roots: • Δ > 0: Two distinct real roots (parabola crosses x-axis twice) • Δ = 0: One repeated root (parabola touches x-axis once) • Δ < 0: No real roots (parabola never crosses x-axis)
These roots are the critical points that divide the number line into regions.
\beginaligned x &= (-b ± √(b^2 - 4ac))/(2a) \quad \textwhere Δ = b^2 - 4ac \\ x_1 &= (-b - √(Δ))/(2a), \quad x_2 = (-b + √(Δ))/(2a) \\ Δ > 0 &\Rightarrow \text2 roots, \quad Δ = 0 \Rightarrow \text1 root, \quad Δ < 0 \Rightarrow \text0 roots \endaligned
Parabola Direction and Shape
The leading coefficient a determines the parabola's direction:
**If a > 0 (positive):** • Parabola opens UPWARD (∪ shape) • Vertex is the minimum point • Arms extend upward to infinity • For > or ≥: solution is typically OUTSIDE the roots • For < or ≤: solution is typically BETWEEN the roots
**If a < 0 (negative):** • Parabola opens DOWNWARD (∩ shape) • Vertex is the maximum point • Arms extend downward to negative infinity • For > or ≥: solution is typically BETWEEN the roots • For < or ≤: solution is typically OUTSIDE the roots
\beginaligned y &= ax^2 + bx + c \quad \textwith vertex at (-(b)/(2a), f(-(b)/(2a))) \\ a > 0 &: \text Opens upward \cup, \quad \lim_x \to ±∞ y = +∞ \\ a < 0 &: \text Opens downward \cap, \quad \lim_x \to ±∞ y = -∞ \endaligned
Step-by-Step Solution Process
**Step 1: Write the inequality in standard form** Ensure it's in the form ax^2 + bx + c \lessgtr 0
**Step 2: Find the critical points (roots)** Solve ax^2 + bx + c = 0 using the quadratic formula
**Step 3: Determine parabola direction** Check if a > 0 (opens up) or a < 0 (opens down)
**Step 4: Identify solution regions** Use the parabola's shape to determine where it satisfies the inequality
**Step 5: Write the solution** Express in interval notation or inequality form
**Step 6: Verify with test points** Test values in each region to confirm the solution
\beginaligned \textStep 1: & \quad ax^2 + bx + c \lessgtr 0 \quad \text(standard form) \\ \textStep 2: & \quad x = (-b ± √(b^2-4ac))/(2a) \quad \text(find roots) \\ \textStep 3: & \quad \textsign(a) arrow \textparabola direction \\ \textStep 4: & \quad \textTest regions: (-∞, x_1), (x_1, x_2), (x_2, ∞) \endaligned
Two Distinct Roots Case (Δ > 0)
When the discriminant is positive, the parabola crosses the x-axis at two points x_1 and x_2 (where x_1 < x_2). These roots divide the number line into three regions:
**Region 1:** x < x_1 (left of both roots) **Region 2:** x_1 < x < x_2 (between the roots) **Region 3:** x > x_2 (right of both roots)
**For parabola opening UP (a > 0):** • ax^2 + bx + c > 0: Solution is x < x_1 OR x > x_2 (outside) • ax^2 + bx + c < 0: Solution is x_1 < x < x_2 (between)
**For parabola opening DOWN (a < 0):** • ax^2 + bx + c > 0: Solution is x_1 < x < x_2 (between) • ax^2 + bx + c < 0: Solution is x < x_1 OR x > x_2 (outside)
\beginaligned Δ > 0 &\Rightarrow x_1 = (-b - √(Δ))/(2a), \quad x_2 = (-b + √(Δ))/(2a) \\ a > 0: & \quad ax^2+bx+c > 0 \Leftrightarrow x \in (-∞, x_1) \cup (x_2, ∞) \\ & \quad ax^2+bx+c < 0 \Leftrightarrow x \in (x_1, x_2) \endaligned
One Repeated Root Case (Δ = 0)
When the discriminant equals zero, the parabola touches the x-axis at exactly one point (the vertex). This creates a special case:
**For parabola opening UP (a > 0):** • ax^2 + bx + c > 0: All x except the root • ax^2 + bx + c ≥ 0: All real numbers • ax^2 + bx + c < 0: No solution • ax^2 + bx + c ≤ 0: Only at the root
**For parabola opening DOWN (a < 0):** • ax^2 + bx + c > 0: No solution • ax^2 + bx + c ≥ 0: Only at the root • ax^2 + bx + c < 0: All x except the root • ax^2 + bx + c ≤ 0: All real numbers
\beginaligned Δ = 0 &\Rightarrow x_0 = -(b)/(2a) \quad \text(double root) \\ a > 0: & \quad ax^2+bx+c ≥ 0 \; \forall x, \quad ax^2+bx+c = 0 \Leftrightarrow x = x_0 \\ a < 0: & \quad ax^2+bx+c ≤ 0 \; \forall x, \quad ax^2+bx+c = 0 \Leftrightarrow x = x_0 \endaligned
No Real Roots Case (Δ < 0)
When the discriminant is negative, the parabola never crosses the x-axis. The entire parabola is either above or below the x-axis:
**For parabola opening UP (a > 0):** • Entire parabola is ABOVE x-axis • ax^2 + bx + c > 0 or ≥ 0: All real numbers • ax^2 + bx + c < 0 or ≤ 0: No solution
**For parabola opening DOWN (a < 0):** • Entire parabola is BELOW x-axis • ax^2 + bx + c > 0 or ≥ 0: No solution • ax^2 + bx + c < 0 or ≤ 0: All real numbers
\beginaligned Δ < 0: & \quad \textNo real roots, parabola does not cross x-axis \\ a > 0: & \quad ax^2+bx+c > 0 \; \forall x \in \mathbbR \quad \text(always positive) \\ a < 0: & \quad ax^2+bx+c < 0 \; \forall x \in \mathbbR \quad \text(always negative) \endaligned
Inequality Symbols and Boundary Points
**Strict Inequalities (< and >):** • Roots are NOT included in the solution • Use open circles (○) at the roots on a number line • Use parentheses ( ) in interval notation
**Non-Strict Inequalities (≤ and ≥):** • Roots ARE included in the solution • Use closed circles (●) at the roots on a number line • Use brackets [ ] in interval notation
Example: • x^2 - 4 > 0: Solution is x < -2 or x > 2, written as (-∞, -2) \cup (2, ∞) • x^2 - 4 ≥ 0: Solution is x ≤ -2 or x ≥ 2, written as (-∞, -2] \cup [2, ∞)
\beginaligned \textStrict: & \quad x > a: (a, ∞), \quad x < a: (-∞, a) \quad \text(open ○) \\ \textNon-strict: & \quad x ≥ a: [a, ∞), \quad x ≤ a: (-∞, a] \quad \text(closed ●) \\ \textExample: & \quad x^2 - 4 > 0 \Rightarrow (-∞, -2) \cup (2, ∞) \endaligned
Interval Notation for Solutions
Solutions to quadratic inequalities are expressed using interval notation:
**Single Interval (between roots):** • (x_1, x_2): between roots, not including them • [x_1, x_2]: between roots, including them • (x_1, x_2]: between roots, including only x_2
**Union of Intervals (outside roots):** • (-∞, x_1) \cup (x_2, ∞): outside roots, not including them • (-∞, x_1] \cup [x_2, ∞): outside roots, including them
**All Real Numbers:** • (-∞, ∞) or \mathbbR
**Empty Set:** • \emptyset or \\ (no solution)
\beginaligned \textBetween: & \quad x_1 < x < x_2 \Leftrightarrow (x_1, x_2), \quad x_1 ≤ x ≤ x_2 \Leftrightarrow [x_1, x_2] \\ \textOutside: & \quad x < x_1 \text or x > x_2 \Leftrightarrow (-∞, x_1) \cup (x_2, ∞) \\ \textSpecial: & \quad \mathbbR = (-∞, ∞), \quad \emptyset = \textno solution \endaligned
Test Point Method
To verify your solution, use the test point method:
1. **Identify regions:** The roots divide the number line into regions 2. **Choose test points:** Pick one value from each region 3. **Evaluate:** Substitute each test point into the original inequality 4. **Determine solution:** Regions where the test point satisfies the inequality are part of the solution
Example for x^2 - 5x + 6 > 0 with roots at x = 2 and x = 3: • Test x = 0: (0)^2 - 5(0) + 6 = 6 > 0 ✓ (Region 1 included) • Test x = 2.5: (2.5)^2 - 5(2.5) + 6 = -0.25 < 0 ✗ (Region 2 excluded) • Test x = 4: (4)^2 - 5(4) + 6 = 2 > 0 ✓ (Region 3 included)
\beginaligned \textFor ax^2+bx+c \lessgtr 0 \text with roots x_1, x_2: \\ \textTest x_t \in (-∞, x_1): & \quad ax_t^2 + bx_t + c \lessgtr 0 \text ? \\ \textTest x_t \in (x_1, x_2): & \quad ax_t^2 + bx_t + c \lessgtr 0 \text ? \\ \textTest x_t \in (x_2, ∞): & \quad ax_t^2 + bx_t + c \lessgtr 0 \text ? \endaligned
Real-World Applications
Quadratic inequalities appear in many practical scenarios:
• **Physics:** Projectile motion - finding when an object is above a certain height • **Business:** Profit analysis - determining production levels for positive profit • **Engineering:** Stress analysis - ensuring materials stay within safe limits • **Economics:** Supply and demand - finding equilibrium price ranges • **Architecture:** Parabolic arches - calculating load distribution • **Sports:** Trajectory analysis - determining if a ball clears an obstacle • **Finance:** Investment returns - identifying profitable investment periods • **Biology:** Population dynamics - modeling sustainable population ranges
\beginaligned \textProjectile: & \quad h(t) = -16t^2 + v_0t + h_0 > h_\texttarget \\ \textProfit: & \quad P(x) = -ax^2 + bx - c > 0 \quad \text(break-even points) \\ \textPhysics: & \quad E_\textkinetic = (1)/(2)mv^2 ≥ E_\textmin \endaligned
Common Mistakes to Avoid
• **Forgetting to check the sign of a:** The parabola direction completely changes the solution • **Misinterpreting strict vs non-strict inequalities:** Remember whether to include or exclude roots • **Ignoring the discriminant:** Always check if roots are real before proceeding • **Incorrect interval notation:** Union (∪) for "OR", not intersection (∩) • **Not testing points:** Always verify your solution with test points • **Confusing "between" and "outside":** Depends on both the parabola direction and inequality type • **Algebraic errors:** Be careful when manipulating inequalities • **Wrong interval brackets:** Use ( ) for open, [ ] for closed
\beginaligned \textCheck: & \quad \textsign(a) arrow \textdirection, \quad Δ = b^2-4ac arrow \textroot count \\ \textVerify: & \quad \textTest points in each region \\ \textNotation: & \quad (a,b) \cup (c,d) \text for OR, \quad [a,b] \text for closed \endaligned
Graphing Quadratic Inequalities Calculator Worked Examples
Worked Example
Inputs
- a: 1
- b: -3
- c: 2
- inequality: >
Result: x < 1 or x > 2
Explanation
x² - 3x + 2 > 0 has roots at x = 1 and x = 2, solution is outside these values
Second Scenario
Inputs
- a: 1.2
- b: -3
- c: 2
- inequality: >
Result: x < 1 or x > 2
Explanation
This scenario uses different inputs (a = 1.2, b = -3, c = 2, inequality = >) to show how changing one variable affects the graphing quadratic inequalities result. Run the calculator above with these values to get the exact updated output with step-by-step work.
Common Graphing Quadratic Inequalities Calculator Use Cases
- Homework and exam practice
- Engineering and science coursework
- Quick verification of hand calculations
- Graphing Quadratic Inequalities homework and study
- Graphing Quadratic Inequalities design and analysis
Graphing Quadratic Inequalities Calculator FAQs
How do I know if my quadratic inequality solution is "between" or "outside" the roots?
The solution pattern depends on both the leading coefficient a and the inequality type. For an upward-opening parabola (a > 0): > or ≥ means OUTSIDE the roots (parabola is above x-axis outside), while < or ≤ means BETWEEN the roots (parabola is below x-axis between). For a downward-opening parabola (a < 0), it's reversed: > or ≥ means BETWEEN, and < or ≤ means OUTSIDE. Always visualize the parabola or use test points to verify.
What should I do when the discriminant is negative (no real roots)?
When Δ = b^2 - 4ac < 0, the parabola never crosses the x-axis. If a > 0 (opens up), the entire parabola is above the x-axis, so ax^2 + bx + c > 0 is true for ALL real numbers, and ax^2 + bx + c < 0 has NO solution. If a < 0 (opens down), the entire parabola is below the x-axis, so ax^2 + bx + c < 0 is true for ALL real numbers, and ax^2 + bx + c > 0 has NO solution. This is actually the simplest case!
What's the difference between $>$ and $\geq$ in quadratic inequalities?
The difference is whether the boundary points (roots) are included in the solution. For > (strict inequality), the roots are NOT part of the solution - use open circles (○) or parentheses ( ) in interval notation. For ≥ (non-strict inequality), the roots ARE included - use closed circles (●) or brackets [ ] in interval notation. For example: x^2 - 4 > 0 gives (-∞, -2) \cup (2, ∞), while x^2 - 4 ≥ 0 gives (-∞, -2] \cup [2, ∞).
How do I handle the special case when the discriminant equals zero?
When Δ = 0, there's exactly one root (a repeated root where the parabola touches but doesn't cross the x-axis). For a > 0: > means all x except the root, ≥ means all real numbers, < means no solution, ≤ means only at the root. For a < 0: > means no solution, ≥ means only at the root, < means all x except the root, ≤ means all real numbers. The key is recognizing that the parabola only touches the x-axis at one point, never dipping below (if a > 0) or above (if a < 0).
Why is checking the sign of the leading coefficient $a$ so important?
The sign of a completely determines the parabola's direction and fundamentally changes the solution. If a > 0, the parabola opens UPWARD (∪), making the region between roots negative and outside regions positive. If a < 0, the parabola opens DOWNWARD (∩), making the region between roots positive and outside regions negative. Forgetting to check this is the #1 most common error - you could get the complete opposite of the correct solution! Always check: "Is my parabola a smile (a > 0) or a frown (a < 0)?"
How do I use test points to verify my solution?
After finding the roots, they divide the number line into regions. Pick any convenient value from each region (often 0 works well for the middle region) and substitute it into the original inequality. If it satisfies the inequality, that entire region is part of the solution. For example, with x^2 - 5x + 6 > 0 (roots at 2 and 3): test x=0: 6 > 0 ✓, test x=2.5: -0.25 > 0 ✗, test x=4: 2 > 0 ✓. So the solution is x < 2 or x > 3.
What is interval notation and why do we use unions (∪)?
Interval notation is a mathematical way to express ranges of values. For quadratic inequalities with two roots, we often have two separate solution regions that don't connect, so we use the union symbol (∪) to combine them. For example, x < -2 or x > 3 is written as (-∞, -2) \cup (3, ∞). The ∪ means "union" or "OR" - values can be in either interval. Don't confuse this with intersection (∩) which means "AND" - that would require values to be in both intervals simultaneously, which is impossible for separate intervals.
Can a quadratic inequality have no solution or be true for all real numbers?
Yes! When the discriminant is negative (Δ < 0), the parabola never crosses the x-axis. If it opens upward (a > 0) and you're looking for where it's less than zero (< 0), there's NO solution because the parabola is always above the x-axis. Conversely, it's > 0 for ALL real numbers. The opposite occurs when a < 0. Additionally, when Δ = 0 with certain inequality types, you can get solutions like "all real numbers" or "no solution" depending on the direction and inequality type.
What are real-world applications of quadratic inequalities?
Quadratic inequalities model many real situations: (1) **Projectile motion** - finding when a ball is above a certain height: -16t^2 + v_0t + h_0 > h_target; (2) **Profit analysis** - determining production levels where profit is positive; (3) **Engineering** - ensuring stress on materials stays within safe limits; (4) **Economics** - finding price ranges where supply exceeds demand; (5) **Architecture** - calculating safe load distributions on parabolic structures; (6) **Sports** - determining if a shot clears an obstacle. Any scenario with a quadratic relationship and threshold constraints uses quadratic inequalities.
How do I convert between different forms of expressing the solution?
Solutions can be expressed three ways: (1) **Inequality form**: x < -2 or x > 3; (2) **Interval notation**: (-∞, -2) \cup (3, ∞); (3) **Set-builder notation**: \x \in \mathbbR \mid x < -2 \text or x > 3\. To convert: identify the boundary points and whether they're included ([ ] for included, ( ) for excluded), determine if solutions are between or outside roots, use ∪ for "OR" (separate regions) and remember that -∞ and +∞ always use parentheses. Each form has its uses: inequalities are intuitive, interval notation is compact, and set-builder notation is formal.