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Quadratic Inequality Calculator

Solve ax² + bx + c ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Quadratic. Sign of a is the outside of the roots. Not cubic. Not a second engine. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Core
Declared partition coverage
PASS · 14/14 declared partitions (inequality-two-roots, inequality-repeated, inequality-no-real-roots, inequality-rhs, inequality-fractions, two-real, repeated, complex, irrational, near-double-root, large-scale, fraction-input, cancellation, invalid-domain) · Matrix
Numerical scope
≤4 ULP vs O3 applies only to the published tabulated vectors (two-real, repeated, complex, irrational, near-double-root, tiny-complex, tiny-real, tiny-scale, fraction-numeric, downward, large-scale, cancellation). It is not a guarantee over the entire coefficient space. Roots use a cancellation-safe q-form at the same ± order as x = (−b ± √Δ)/(2a).
Known limitations
  • Declared partitions are two-real, repeated, complex, irrational, near-double-root, large-scale, fraction-input, cancellation, invalid-domain — plus inequality-two-roots, inequality-repeated, inequality-no-real-roots, inequality-rhs, inequality-fractions; not a claim that every quadratic is covered. Near-double-root, large-scale, and cancellation are numerical-boundary partitions and are in the O3 tabulated ULP claim.
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
The solution set of ax² + bx + c ○ rhs on ℝ for ○ in {<,≤,>,≥}, with a parabola Live graph and an engine-linked number line.
Scope
Degree 2 only
Verification
Engine tested · Source checked · v1.1.1 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.1 · CVP protocol 1.0.0-proposed · Evidence 2026-10-13.quadratic-interface-replay
Verification revision
2026-10-13.quadratic-interface-replay · 5/5 property · digest 4dddb83bf6df
Legacy regression
9/9 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · Overall CVP state: VERIFIED_STALE under CVP 1.0.0-proposed
CVP status
STALE · attestation snapshot core d90797326c81 ≠ live cvp_core_sha256 4dddb83bf6df (2026-10-13.quadratic-interface-replay)
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 3/3 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Core profile does not include Live-graph integration. Label collision, near-double-root markers, viewport switching, and pointer interaction need independent visual tests.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
40/40 golden · 3/3 CVP boundary · 10/10 invalid · 5/5 property · 1/1 metamorphic · 12/12 O3 · 11/11 cross-interface · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
6 legacy golden · 3 legacy boundary · legacy regression suite · 40/40 oracle-backed golden · 10/10 invalid · Artifact integrity PASS
STALE · Build schema 1.1.1 matches page · production binding STALE · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Equality rootsΔ = b² − 4a(c − rhs)
a > 0, f < 0(lo, hi)
a > 0, f > 0(−∞, lo) ∪ (hi, ∞)
iSame engine as /quadratic. Default rhs is 0. Not cubic. Not CAS factoring. Linear inequalities stay at /linear-inequality.

How to use

1

Enter a, b, c, ○, and rhs

x² − 5x + 6 < 0 is a=1, b=−5, c=6, op=<, rhs=0. The equation page stays at /quadratic.

2

Read the solution set

(2, 3), or ℝ / ∅ / a union of rays. The Live graph is the parabola; the number line is the published interval. Cubic inequalities are out of this engine.

Example calculations

Common configurations with formula and result.

ϟ

Between the roots

x² − 5x + 6 < 0

(x − 2)(x − 3) < 0
(2, 3)
ϟ

Outside the roots

x² − 5x + 6 > 0

opens up
(-∞, 2) ∪ (3, ∞)
ϟ

Always true

x² + 1 > 0

Δ < 0, a > 0
ℝ

Quadratic Inequality calculator specification

Version 1.1.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
ax² + bx + c ○ rhs with ○ in {<,≤,>,≥} is an interval, a union of rays, a point, ℝ, or ∅. This page is mode=inequality on math.quadratic. The sign of a is the outside of the roots. a ≠ 0. Not a second engine. Not cubic. Not CAS.
What it calculates
The solution set of ax² + bx + c ○ rhs on ℝ for ○ in {<,≤,>,≥}, with a parabola Live graph and an engine-linked number line.
Inputs
  • a
  • b
  • c
  • op
  • rhs?
Outputs
  • interval
  • status
  • delta
  • lo
  • hi
Formula
ax² + bx + c ○ rhs
Assumptions
  • Degree 2 only
  • Real coefficients
  • a ≠ 0
  • The Result Card includes the parabola Live graph plus an engine-linked number line of the published interval. Same interval as REST/MCP.
  • Not cubic
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • Non-finite results or discriminant product overflow/underflow → NUMERICAL_RANGE_ERROR; rescale coefficients and rhs
  • missing a, b, or c → MISSING_REQUIRED_INPUT
  • a = 0 → NOT_QUADRATIC
  • unknown op → INVALID_MODE
  • unknown mode (including factor, cas, cubic) → INVALID_MODE
Example
a=1 b=-5 c=6 op=lt → (2, 3)
Validation cases

2 published on this page · 9/9 tests · Production surface contract 6/6 · View evidence

  • a=1 b=-5 c=6 op=lt → (2, 3)
  • mode=factor a=1 b=-5 c=6 → INVALID_MODE
Sources
  • University of Oxford Mathematical Institute — Algebra — Linear/quadratic inequalities and sign of factors
    Supports: Factoring polynomials and determining sign intervals, with explicit quadratic-inequality examples. For a rational inequality the calculator applies the same factor-sign rules to numerator and denominator separately and excludes denominator zeros; that rational extension is a derived application, not a claim that the page tests this solver. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version
1.1.1

Background

Interpretation and common distinctions.

What does this quadratic inequality calculator compute?

ax² + bx + c ○ rhs with ○ in {<,≤,>,≥} is an interval, a union of rays, a point, ℝ, or ∅.

The published formula or method is ax² + bx + c ○ rhs. The following references support the stated mathematics or method:

  • University of Oxford Mathematical Institute — Algebra: Factoring polynomials and determining sign intervals, with explicit quadratic-inequality examples. For a rational inequality the calculator applies the same factor-sign rules to numerator and denominator separately and excludes denominator zeros; that rational extension is a derived application, not a claim that the page tests this solver. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

The discriminant determines real and complex root cases. An inequality additionally requires the correct sign intervals and endpoint inclusion.

The worked example uses x² − 5x + 6 < 0 with (x − 2)(x − 3) < 0. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.quadratic v1.1.1 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Absolute Value Calculator, Absolute Value Inequality Calculator, Cubic Equation Calculator, Linear Equation Calculator, Linear Inequality Calculator, Polynomial Evaluate Calculator, Polynomial Roots Calculator, Quadratic Formula Calculator . Explore all Algebra & Equations.

→ Absolute Value Calculator Solve |ax + b| = rhs. Same engine as Linear Equation. Two solutions when rhs > 0; empty when rhs < 0. Not quadratic. Not a second engine. Not CAS. Runs locally. › → Absolute Value Inequality Calculator Solve |ax + b| ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Linear Equation. Inside the branches is an interval; outside is a union. Not quadratic. Not a second engine. Not CAS. Runs locally. › → Cubic Equation Calculator Cardano / trigonometric closed form for a real cubic. Same engine as Polynomial Roots. Three real roots or one real plus a conjugate pair. Not Ferrari. Not a second cubic engine. Runs locally. › → Linear Equation Calculator Solve ax + b = rhs for x. Identity and inconsistent cases when a = 0. Inequality /linear-inequality. Absolute value /absolute-value. Not the quadratic formula. Runs locally. › → Linear Inequality Calculator Solve ax + b ○ rhs for ○ in {<, ≤, >, ≥}. Same engine as Linear Equation. Negative a flips the inequality. Not quadratic. Not a second engine. Runs locally. › → Polynomial Evaluate Calculator Horner nested multiplication for a real polynomial at a real x. Same engine as Polynomial Roots. Not factoring. Not a second engine. Not CAS. Runs locally. › → Polynomial Roots Calculator Find all roots of a real polynomial of degree 1–8. Closed form for linear, quadratic, and cubic (Cardano); Durand–Kerner otherwise. Cubic discovery /cubic-equation. Horner evaluation /polynomial-evaluate. Not the quadratic formula page. Runs locally. › → Quadratic Formula Calculator Solve ax² + bx + c = 0 with the quadratic formula. Inequality discovery /quadratic-inequality. Free online tool — no sign-up. Clear formulas and worked examples. Runs locally in your browser. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second quadratic engine?

No. It is mode=inequality on math.quadratic, the same engine as /calc/math/quadratic.

Why does negative a flip the set?

The parabola opens downward, so the sign of f is the opposite of the a>0 case. The engine uses the sign of a as the outside of the roots.

Do you solve cubic inequalities?

No. Degree 3 stays on /calc/math/cubic-equation for roots. A cubic-inequality discovery would be a later Entry on math.polynomial.roots, not this page.