2026-09-29 9 min

Quadratic Inequality Calculator

Free quadratic inequality calculator: solve ax² + bx + c <, <=, >, >= 0 with discriminant, roots, sign chart and interval notation.

Calculator

Quadratic Coefficients

Coefficient a

Coefficient b

Constant c

Inequality operator

Results update live as you type

Status:1x² + -5x + 6 < 0(2, 3)

Solution Summary

Inequality1x² + -5x + 6 < 0

DiscriminantD = 1

Solution(2, 3)

Results

Optimal

Solution set

(2, 3)

Union of the matching sign regions

Discriminant D

b² - 4ac; negative means no real roots

Vertex

(2.5, -0.25)

Parabola opens upward

Roots

2 and 3

Sign changes only here

Results are recomputed live and verified against the step-by-step work below.

Visualizations

Parabola key values

-0.2Vertex y

0f(r1 area)

6y-intercept

Real roots

Two distinct roots100%

Formula

A quadratic keeps one sign between consecutive real roots, so the roots plus the opening direction determine everything.

ax² + bx + c ? 0, D = b² - 4ac

Alternative Form

Roots x = (-b ± √D) / 2a split the line into at most three sign regions

Variables

Symbol	Name	Unit	Description
a	Leading coeff a	number	Nonzero; sign sets opening direction
b	Linear coeff b	number	Controls root positions
c	Constant c	number	y-intercept
D	Discriminant	number	b² - 4ac; counts real roots

Explanation

Compute the discriminant, find the real roots, then keep the regions whose sign matches the operator; a negative discriminant means the sign never changes.

Derivation

Factoring (where possible) as a(x-r1)(x-r2) shows each factor flips sign at its root; the product sign per region follows directly.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write the inequality.

ax² + bx + c ? 0

stmt:1x² + -5x + 6 < 0

Result: Statement recorded

Step 2

Compute the discriminant.

D = b² - 4ac

D:1

Intermediate: -5² - 4(1)(6) = 1

Result: D = 1

Step 3

Two roots split the line into three regions.

x = (-b ± √D) / 2a

roots:2, 3

Result: 2 real root(s)

Step 4

Opens upward; keep regions whose sign matches < 0.

Sign chart

iv:(2, 3)

Result: (2, 3)

Step 5

Write the matching regions in interval notation.

Solution

iv:(2, 3)

Result: (2, 3)

Final Answer

(2, 3)

Worked Examples

See how the calculator works with different scenarios.

Scenario

Upward parabola, between the roots.

Inputs

a:1

b:-5

c:6

op:<

Calculation

D = 1, roots 2 and 3, negative between them

Result

(2, 3)

Interpretation

Only the middle region dips below the axis.

Understanding Your Result

What This Means

The headline result is the exact answer for the quadratic inequality calculator inputs you entered, with supporting values shown beside it.

Why It Matters

Homework, exams and real-world models all reduce to this calculation; a verified numeric answer beats guessing.

Professional Interpretation

Read the headline result first, then scan the supporting rows and the step list to confirm the arithmetic before reusing the value.

Recommendations

•Verify the headline result against the steps
•Test one worked example by hand
•Check edge cases separately
•Keep full precision until the final display
•Restate the answer in the problem context

Educational Content

Introduction

This quadratic inequality calculator evaluates your inputs with closed-form arithmetic and shows every intermediate value.

Concept Explanation

Compute the discriminant, find the real roots, then keep the regions whose sign matches the operator; a negative discriminant means the sign never changes.

Applications

▸Homework and exams

▸Engineering checks

▸Finance models

▸Science labs

▸Programming logic

▸Everyday estimation

Benefits

•Instant verified answer
•Step-by-step work shown
•Charts that expose mistakes
•Handles decimals directly
•Free with examples

Limitations

•Numeric inputs only
•Real-number results
•Display rounding to 6 decimals

Best Practices

•Enter numbers carefully
•Check the steps line by line
•Verify with one example
•Watch units in word problems
•State the final answer with context

Common Misconceptions

Myth: One input is enough.

Truth: Most problems need every coefficient; missing values change the answer.

Myth: Rounding early is fine.

Truth: Keep full precision until the final display to avoid drift.

Myth: The chart is decoration.

Truth: The chart is a second computation — mismatched bars mean a bad input.

Myth: Any rearrangement works.

Truth: Only equivalent operations preserve equality; follow the steps.

Practical Tips

•Double-check each input
•Read the steps in order
•Compare against one worked example
•Verify edge cases separately
•Keep units consistent
•Use Reset to restore defaults
•Copy the summary for notes

Common Mistakes

Typing a value in the wrong field

Match each label to the formula variables before calculating.

Dividing by zero

Check denominators and leading coefficients first.

Ignoring the steps

The step list shows exactly where a sign or arithmetic slip hides.

Rounding mid-calculation

Round only the displayed answer.

Forgetting to verify

Substitute the answer back into the original statement.

Assumptions

This calculator makes the following assumptions to provide estimates:

Real numbers: Inputs and outputs are real numbers.

Exact inputs: Coefficients are taken as given.

Standard notation: Conventional symbols and operator precedence.

Display rounding: Six decimals in outputs.

Limitations

Be aware of these limitations when using this calculator:

Numeric scope: Symbolic parameters need the advanced pages.

Single scenario: One input set per calculation.

No units conversion: Convert units before entering values.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I solve a quadratic inequality?

Compute D = b² - 4ac, find the real roots, then keep the regions whose sign matches the operator. For x² - 5x + 6 < 0 the roots 2 and 3 give (2, 3).

What does the discriminant tell me?

D > 0 means two roots and up to three regions; D = 0 means one touching point; D < 0 means the sign never changes.

Does the opening direction matter?

Yes — with the same roots, an upward parabola is negative between them while a downward one is positive there.

When do I use brackets?

Only with <= and >=, which include the roots themselves.

What if D is negative?

Then the answer is either everything or nothing, depending on whether the constant sign matches the operator.

How do I verify?

Test one point from each region in the original inequality.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Showing your steps

Write one operation per line.

Checking answers

Substitute back every time.

Reading interval notation

Brackets include, parentheses exclude.

Word-problem setup

Name variables before computing.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Solution set: All values satisfying the given statement.
Interval notation: Compact bracket notation for ranges of the number line.
Critical value: A point where the expression is zero or undefined, splitting sign regions.
Verification: Substituting the answer back to confirm the original statement.
Closed form: An exact formula evaluated directly, not an approximation loop.
Edge case: Inputs where the usual rule breaks, such as zero denominators.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Math foundations Math is Fun — Index Purplemath — Lessons Wikipedia — Mathematics Cuemath — Math resources

Applicable Standards

Common Core HSA — creating and reasoning with equationsNCTM Algebra — representations and equivalence

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.