HomeCalculatorsMathAlgebra & EquationsQuadratic Formula Calculator
Math calculator

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.

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
Roots of ax²+bx+c=0 via the quadratic formula, with discriminant, vertex, axis of symmetry, optional factorization, and a live parabola of y = ax²+bx+c.
Scope
a ≠ 0; coefficients are real numbers (fractions allowed).
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.

Equationax² + bx + c = 0 (a ≠ 0)
Quadratic formulax = (−b ± √(b² − 4ac)) / (2a)
DiscriminantΔ = b² − 4ac
Vertexxᵥ = −b/(2a), yᵥ = f(xᵥ)
iΔ > 0 → two distinct real roots; Δ = 0 → one repeated real root; Δ < 0 → two complex roots. Fractions such as 3/4 are accepted for a, b, c.

How to use

1

Enter coefficients a, b, c

From ax² + bx + c = 0. a must be nonzero. Decimals or fractions (3/4) work.

2

Read the roots

See x₁ / x₂, discriminant Δ, vertex, axis, and a live parabola of y = ax²+bx+c.

3

Follow the steps

The work panel expands the quadratic formula with normalized coefficients.

Example calculations

Common configurations with formula and result.

ϟ

Two real roots

x² − 6x + 8 = 0

Δ = 4
x = 2, x = 4
ϟ

Repeated root

x² − 4x + 4 = 0

Δ = 0
x = 2
ϟ

Complex roots

x² + 2x + 5 = 0

Δ = −16
x = −1 ± 2i
ϟ

Irrational real roots

x² − 8x + 5 = 0

4 ± √11
≈ 7.317, 0.683

Discriminant cases

Common values at a glance.

ΔRoots
Δ > 0Two distinct real roots
Δ = 0One repeated real root
Δ < 0Two complex conjugate roots
i Example: x² − 6x + 8 = 0 → Δ = 4 → x = 2 and x = 4. x² + 2x + 5 = 0 → Δ = −16 → x = −1 ± 2i.

Quadratic Formula calculator specification

Version 1.1.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
A quadratic equation is a second-degree polynomial ax² + bx + c = 0 with a ≠ 0. Its solutions (roots) are given by the quadratic formula x = (−b ± √(b² − 4ac)) / (2a). The discriminant Δ = b² − 4ac tells whether the roots are two real, one repeated real, or a complex conjugate pair.
What it calculates
Roots of ax²+bx+c=0 via the quadratic formula, with discriminant, vertex, axis of symmetry, optional factorization, and a live parabola of y = ax²+bx+c.
Inputs
  • Quadratic coefficient a ≠ 0
  • Linear coefficient b
  • Constant c
Outputs
  • x₁ and x₂ (real or complex conjugate pair)
  • Discriminant Δ and root type
  • Vertex (xᵥ, yᵥ), axis x = xᵥ, y-intercept, opens up/down
  • Live parabola with root, vertex, and intercept markers
  • Factor form when roots are small rationals
Formula
x = (−b ± √Δ)/(2a), Δ = b²−4ac
Assumptions
  • a ≠ 0; coefficients are real numbers (fractions allowed).
Units
  • Dimensionless (or consistent units of x)
Boundary conditions
  • Non-finite results or discriminant product overflow/underflow → NUMERICAL_RANGE_ERROR; rescale coefficients and rhs
  • a = 0 → rejected as not quadratic
Example
x² − 5x + 6 = 0 → x = 2, x = 3
Validation cases

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

  • a=1,b=-6,c=8 → 2, 4
  • a=1,b=-4,c=4 → 2
  • a=1,b=2,c=5 → −1 ± 2i
  • a=1,b=-8,c=5 → 4 ± √11
Sources
  • Margaret Stevens — The Maths Workbook (University of Oxford) — Chapter 1, §3.4: Quadratic Equations
    Supports: The quadratic formula, discriminant and real-root multiplicity. Negative discriminants have no real roots; the calculator’s complex-root representation uses the separately cited complex-number definitions. This does not validate the numerical cancellation-avoiding algorithm. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
  • Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare) — §9.4 Complex Numbers
    Supports: Complex arithmetic, conjugates, modulus and argument, polar/exponential multiplication and division, de Moivre’s formula and complex roots. Noninteger powers require a specified branch; those branch and numerical policies remain CalculatorX’s own contract. 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 formula calculator compute?

A quadratic equation is a second-degree polynomial ax² + bx + c = 0 with a ≠ 0. Its solutions (roots) are given by the quadratic formula x = (−b ± √(b² − 4ac)) / (2a). The discriminant Δ = b² − 4ac tells whether the roots are two real, one repeated real, or a complex conjugate pair.

The published formula or method is x = (−b ± √Δ)/(2a), Δ = b²−4ac. The following references support the stated mathematics or method:

  • Margaret Stevens — The Maths Workbook (University of Oxford): The quadratic formula, discriminant and real-root multiplicity. Negative discriminants have no real roots; the calculator’s complex-root representation uses the separately cited complex-number definitions. This does not validate the numerical cancellation-avoiding algorithm. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
  • Gilbert Strang, Calculus — Chapter 9: Polar Coordinates and Complex Numbers (MIT OpenCourseWare): Complex arithmetic, conjugates, modulus and argument, polar/exponential multiplication and division, de Moivre’s formula and complex roots. Noninteger powers require a specified branch; those branch and numerical policies remain CalculatorX’s own contract. 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 − 6x + 8 = 0 with Δ = 4. 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.

What is a quadratic equation?

A quadratic equation is a second-degree polynomial:

ax² + bx + c = 0 (a ≠ 0)

  • a — quadratic coefficient
  • b — linear coefficient
  • c — constant term

The solutions are the roots (where the parabola y = ax²+bx+c meets the x-axis, when real).

Quadratic formula

x = (−b ± √(b² − 4ac)) / (2a)

Discriminant

Δ = b² − 4ac

Δ Meaning
> 0 Two distinct real roots
= 0 One repeated real root
< 0 Two complex conjugate roots

Vertex and axis

xᵥ = −b/(2a), yᵥ = a xᵥ² + b xᵥ + c

The axis of symmetry is the vertical line x = xᵥ.

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 Inequality 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 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. ›

Frequently asked questions

Key distinctions behind the calculation.

What is a quadratic equation?

A polynomial equation of degree two: ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola.

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) / (2a). It gives the roots for any quadratic with a ≠ 0.

What does the discriminant tell me?

Δ = b² − 4ac. Positive → two real roots; zero → one repeated root; negative → two complex roots.

What if a = 0?

Then the equation is linear (bx + c = 0), not quadratic. This calculator requires a ≠ 0.

What is the vertex?

The turning point of the parabola. xᵥ = −b/(2a) is also the axis of symmetry; yᵥ = a xᵥ² + b xᵥ + c.

Are complex roots allowed?

Yes. When Δ < 0 the roots are written as real ± imaginary i.

Do you solve quadratic inequalities?

Yes. /calc/math/quadratic-inequality is mode=inequality on this same engine. Linear inequalities stay at /linear-inequality.