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Quadratic Formula Calculator

Solve ax² + bx + c = 0 with the quadratic formula. 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 · 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
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 · v1.0.4 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.4 · 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 07d7671fb2dd
Legacy regression
9/9 tests · Production surface contract 4/4
Trust layers
Verification VERIFIED · Production CURRENT · overall VERIFIED
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
This calculator CURRENT · Public schema 1.0.4 matches · Semantic contract ✓ · Production attested · 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 1.0.4 · 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
  • 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 4/4 · 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
Calculation version
1.0.4

Background

Interpretation and common distinctions.

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: Exponent, Square root, Factorial, GCF .

x^y Exponent › √ Square root › n! Factorial › gcd GCF ›
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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.

Can I enter fractions?

Yes. Values like 3/4 or -3/2 are accepted for a, b, and c.