Quadratic Formula Calculator

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 v1.1.2 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

Vertex and axis

xᵥ = −b/(2a), yᵥ = a xᵥ² + b xᵥ + c

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

Numerical scope and reproducible interpretation

The accepted inputs are finite binary64 real coefficients with a nonzero a. Coefficients accept decimal/scientific notation, en-US grouping, Unicode minus, or an integer fraction. Each text token is limited to 512 characters. Fractions are converted once to binary64; exact decisions below refer to those accepted values. Nonzero text that rounds to zero rejects NUMERIC_UNDERFLOW. Missing, blank or null coefficients remain invalid. Matching aliases are accepted; conflicting aliases, unknown fields and solve-mode inequality inputs are rejected.

The discriminant, vertex and shifted inequality constant c minus rhs use bounded exact rational arithmetic. This prevents rounded subtraction from changing two distinct roots into a repeated root or an empty solution set. A scaled square root and cancellation-resistant q formula produce roots. A required discriminant, root or vertex outside finite binary64, or two distinct roots that collapse to the same binary64 endpoint, rejects NUMERICAL_RANGE_ERROR. Exact arithmetic is limited to 16384 bits. Finite inputs do not guarantee success; rescaling changes rounding and must be considered carefully.

Equation mode reports roots in the plus/minus formula order, the discriminant, vertex, axis and y-intercept. Inequality mode supports lt, le, gt and ge; an omitted right-hand side defaults to zero, while an explicitly blank right-hand side is invalid. The inequality entry locks its mode. Repeated-root inequalities can return a point, a punctured line, all real numbers or the empty set. Close roots can be highly sensitive to coefficient perturbations. Exact branch decisions do not imply exact root coordinates.

Display preserves complete nonzero and distinct binary64 values. Optional factorization appears only when exact coefficient reconstruction succeeds; nearby simple fractions are not substituted for coefficients. Parabola samples use exact rational Horner arithmetic on accepted inputs. Graphs are explicitly omitted when finite coordinates, clusters, tiny features or sampling range cannot be displayed faithfully. Full scalar results, interval endpoints, raw inputs and the typed result pack remain in CSV/JSON exports. CSV text is neutralized against spreadsheet formulas.

The independent reference uses CPython Fraction predicates and Decimal800/1100 square roots, with identical rounded reference results at both precisions. It checks 66 complete result packs, including four inequality operators, negative leading coefficients, repeated/complex roots, fractions, extreme scales, cancellation and raw aliases. The fixed-vector root budget is at most four ULP; other numeric fields, types, topology and presentation are strict. This is not a global error bound or a guarantee of bitwise agreement across runtimes. Historical 1.1.1 verification artifacts remain available; the candidate uses a new complete-pack suite.

Typing computes locally. Share URLs and REST/MCP requests transmit inputs and may enter browser, server or intermediary logs. Old version links warn before current-engine execution; they do not replay a historical runtime. See NIST DLMF polynomial definitions and Python Decimal arithmetic for background; these references are not implementation verification or endorsement.