HomeCalculatorsMathAlgebra & EquationsQuadratic Inequality Calculator
Math calculator

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 · 9/9 declared partitions (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, 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 — 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 {<,≤,>,≥}.
Scope
Degree 2 only
Verification
Engine tested · Source checked · v1.1.0 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-16.sources-no-wolfram
Verification revision
2026-09-16.sources-no-wolfram · 5/5 property · digest 5e416940420d
Legacy regression
9/9 tests · Production surface contract 4/4
Trust layers
Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status
STALE · STALE · Last attested schema matched 1.1.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. 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
9/9 golden · 3/3 CVP boundary · 3/3 invalid · 5/5 property · 1/1 metamorphic · 9/9 O3 · 2/2 cross-interface · 4/4 CVP contract · Manifest
Sources
Sources
Evidence
6 legacy golden · 3 legacy boundary · legacy regression suite · 9/9 oracle-backed golden · 3/3 invalid · Artifact integrity PASS
STALE · Last attested schema matched 1.1.0 snapshot / local build · 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. 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 1.1.0 · 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 {<,≤,>,≥}.
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
  • Not cubic
  • Not CAS
Units
  • dimensionless
Boundary conditions
  • 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 4/4 · View evidence

  • a=1 b=-5 c=6 op=lt → (2, 3)
  • mode=factor a=1 b=-5 c=6 → INVALID_MODE
Sources
Calculation version
1.1.0

Other calculators in this family: Absolute Value Calculator, Absolute Value Inequality Calculator, Column Space Calculator, Condition Number Calculator, Cubic Equation Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear Equation 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. › → Column Space Calculator Column space (image) of a small real square matrix from RREF pivot columns. Same engine as Linear System. Trivial image is {0}. Left-nullspace is a sibling. Runs locally. › → Condition Number Calculator κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix. Same engine as Linear System. Singular matrices return infinite. Not a 2-norm product. 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. › → Least Squares Calculator Linear least squares for a small real square system. Full rank uses QR; rank-deficient uses SVD. Same engine as Linear System. Not Statistics regression. Runs locally. › → Left Nullspace Calculator Left-nullspace (cokernel) of a small real square matrix as ker(Aᵀ). Same engine as Linear System. Trivial left kernel is {0}. Not Jordan form. 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. ›
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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.