quadratic equation
Enable an AI agent to recognise a quadratic equation, determine its admissible solutions, and choose valid transformations and solution methods under explicit domain assumptions.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
recalled by Codex without web access - no source was read
Researched by: Codex
Purpose and description
Enable an AI agent to recognise a quadratic equation, determine its admissible solutions, and choose valid transformations and solution methods under explicit domain assumptions.
A quadratic equation in one variable is a polynomial equation of degree two, conventionally written ax² + bx + c = 0 with a ≠ 0.
It can be Normalise an equation into standard form while preserving domain restrictions.; Classify roots using coefficient assumptions and the discriminant.; Select factoring, completing the square or the quadratic formula when their prerequisites hold.; Partition parameter values into quadratic and degenerate cases.; Compute exact or approximate roots and verify their admissibility in the original equation.; Compare representations and identify whether their solution sets coincide under stated assumptions..
Distinguishing features
After valid simplification, the equation is polynomial in one designated unknown and has degree exactly two.
The leading coefficient is nonzero under the stated parameter assumptions; cases where it vanishes require separate classification.
The equality imposes a constraint to solve, whereas a quadratic expression alone has no asserted equality and a quadratic function specifies a mapping.
Expressions such as xy + x + 1 = 0 are not univariate quadratic equations unless the unknown designation and treatment of other symbols make the degree exactly two.
An equation that becomes quadratic only after squaring, substitution or denominator clearing requires recorded domain restrictions and a solution correspondence.
Scope
+ Recognition of the form ax² + bx + c = 0 with a nonzero
+ Coefficient domain, unknown domain and parameter restrictions
+ Standard, factored and completed-square representations
+ Roots, multiplicities and admissibility constraints
+ Solution methods, equivalence conditions and numerical verification
- Quadratic functions as mappings with their full graph and optimisation behaviour
- Quadratic inequalities and their interval solution sets
- Multivariable quadratic equations, quadratic forms and conic sections
- General polynomial equations of degrees other than two
- Documents, software implementations and physical systems in which an equation occurs
Characteristics
- Designated unknown
- Symbol and interpretation of the single unknown Degree and coefficient status depend on which symbol is being solved for.
- Coefficient triple
- (a, b, c), with exact or approximate values and any applicable units Determines the equation and whether its terms are dimensionally compatible.
- Coefficient domain
- Integers, rationals, reals, complex numbers or another explicitly identified algebraic domain Controls permissible arithmetic, factorisation and applicability of solution formulas.
- Admissible unknown domain
- Specified number system with optional interval, integrality or contextual restrictions A mathematically valid root may fall outside the permitted solution set.
- Quadratic status
- Confirmed quadratic, conditional on parameters, degenerate or unresolved Prevents use of quadratic methods when the leading coefficient can be zero.
- Discriminant
- Δ = b² − 4ac, represented exactly or with numerical uncertainty For real coefficients, its sign distinguishes two distinct real roots, a repeated real root and no real roots.
- Root multiplicity
- Simple or double for each established root Distinguishes a repeated root from two distinct solutions.
- Representation relationship
- Equivalent equation, conditional equivalence or transformation with candidate solutions requiring checks Records whether a rewritten equation preserves the original admissible solution set.
- Solution status
- Unsolved, partially solved, fully solved or numerically unresolved Separates finding candidates from establishing all admissible solutions.
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 6 bundles · 11 layers · 16 findings · 26 questions.
Quadratic identity Establishes what makes the recorded constraint a quadratic equation in one unknown.
An equation's apparent notation can conceal a different degree, a different unknown or a conditional leading coefficient.
Unknown and degree
Identifies the unknown and establishes the degree after justified simplification.
Univariate quadratic test
Recognition requires a designated unknown and a polynomial form of degree exactly two in that unknown.
- Which symbol is the unknown, and which remaining symbols are fixed coefficients or parameters? definition
- After collecting terms through valid operations, is the highest nonzero power of the unknown exactly two? boundary
Coefficient cases
Records the coefficient triple and conditions on the leading coefficient.
Nonzero leading coefficient
The quadratic classification requires a ≠ 0; a = 0 produces a linear or constant equation that must be handled separately.
- What are a, b and c in ax² + bx + c = 0, and are their values exact, approximate or parameter-dependent? measurement
- For which parameter values does a vanish, and what equation remains in each such case? boundary
Domains and admissibility Makes the arithmetic setting and restrictions on acceptable roots explicit.
The same written quadratic can have different solution sets and require different methods in different domains.
Algebraic setting
Separates the coefficient domain from the domain in which solutions are sought.
Permitted arithmetic
Division, square roots and factorisation must be interpreted in the declared algebraic setting; the familiar quadratic formula is not universally applicable.
- In which domain do the coefficients lie, and in which domain must the unknown take values? definition
- Are division by 2a and the required square-root operations available in this setting? action
Inherited restrictions
Preserves restrictions supplied by an original expression or an application.
Admissible root filter
Roots of the polynomial are candidates until checked against exclusions, interval bounds, integrality requirements and contextual constraints.
- Which restrictions on the unknown were inherited from denominators, radicals, substitutions or the originating problem? provenance
- Which polynomial roots satisfy every recorded restriction, including any unit and sign requirements? boundary
Representations and transformations Connects standard, factored and completed-square forms through justified transformations.
Different forms reveal different properties, but transformations can lose restrictions or introduce candidates.
Equivalent forms
Records useful representations and the conditions under which they express the same constraint.
Form correspondence
Standard form exposes coefficients, factored form exposes roots when the factors exist in the chosen domain, and completed-square form exposes a squared offset.
- Which standard, factored or completed-square forms are available in the declared domain? definition
- Does expansion or another valid identity establish that each proposed form preserves the original equation? action
Transformation conditions
Tracks operations whose validity depends on nonzero expressions or restricted substitutions.
Solution-set preservation
Multiplication, division, squaring and changes of unknown require explicit checks for lost roots, introduced roots and excluded values.
- Which steps require a divisor to be nonzero or a substitution to be invertible on the admissible domain? boundary
- How will candidates be mapped back and checked when the quadratic was obtained from a different original equation? action
Root structure and solving Determines root type and multiplicity and selects a justified solution procedure.
An agent must distinguish the existence and structure of roots from the mechanics of computing them.
Discriminant classification
Uses the discriminant within its applicable domain to classify roots.
Real-coefficient root cases
For real coefficients with a ≠ 0, positive, zero and negative discriminants indicate two distinct real roots, a double real root and a nonreal conjugate pair respectively.
- What is b² − 4ac, and is its sign established exactly or only estimated numerically? measurement
- Given the coefficient and solution domains, how many distinct roots exist and what are their multiplicities? definition
Solution method selection
Chooses a method based on available structure and the required output.
Method preconditions
Factoring, direct square-root extraction, completing the square and the quadratic formula offer different routes whose suitability depends on coefficients and domain assumptions.
- Do zero coefficients, a visible factorisation or a perfect-square form permit a simpler justified solution? action
- Should the result be exact, parameter-conditional or numerical, and which applicable method supports that requirement? action
Verification and numerical reliability Checks solution correctness, completeness and sensitivity to numerical error.
A reported root can be inadmissible, incomplete or inaccurate even when a familiar formula was used.
Solution validation
Checks candidates against the original constraint and accounts for all applicable cases.
Correctness and completeness
Validation combines substitution into the original equation with a domain-appropriate argument that all admissible roots and parameter cases have been covered.
- Does each candidate satisfy the original equation exactly or to a justified numerical tolerance? action
- What establishes that no admissible root or exceptional parameter case was omitted? boundary
Finite-precision behaviour
Records precision limits, cancellation risk and sensitivity near repeated roots.
Numerical root confidence
Finite-precision evaluation can suffer cancellation, overflow or underflow, while roots near a repeated-root case can be sensitive to coefficient perturbations; a small residual alone does not establish a small root error.
- What coefficient uncertainty and arithmetic precision affect the discriminant and computed roots? measurement
- Is rescaling, a cancellation-avoiding evaluation or higher precision needed to justify the requested accuracy? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- This describes the conventional algebraic sense in one variable; the supplied domain code INF.MED should be checked for classification fit.
- Root counts and discriminant statements assume real or complex coefficients as specified; other coefficient fields require additional qualifications.
- This is recalled mathematical knowledge, with no sources consulted.
- Which of these check these first hold for the sense of quadratic equation this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Quadratic equations with real coefficients
- Quadratic equations with complex coefficients
- Monic quadratic equations, with a = 1
- Pure quadratic equations, with b = 0
- Quadratic equations with zero constant term, c = 0
- Which of these kinds and varieties hold for the sense of quadratic equation this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Calculating times or positions in constant-acceleration motion.
- Finding intersections of a parabola with a straight line.
- Solving area and dimension constraints in geometry.
- Finding break-even points in quadratic revenue or cost models.
- Determining eigenvalues of a two-by-two matrix through its characteristic equation.
- Which of these real-world use hold for the sense of quadratic equation this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Polynomial degree - Exactly 2 - Dimensionless
- Number of distinct real roots for real coefficients - 0, 1, or 2 - Count
- Number of complex roots counted with multiplicity - Exactly 2 for complex coefficients - Count
- Which of these typical measurements hold for the sense of quadratic equation this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Treating an equation with a = 0 as quadratic; its degree is lower.
- For real coefficients, misclassifying roots: the discriminant b² − 4ac is positive for two distinct real roots, zero for a repeated real root, and negative for two nonreal complex conjugate roots.
- Losing a root by dividing by an expression that can equal zero, such as x when c = 0.
- Introducing extraneous solutions through transformations such as squaring without checking the original equation.
- Losing numerical accuracy through cancellation in direct evaluation of the quadratic formula, especially when subtracting nearly equal quantities.
- Which of these failure modes and hazards hold for the sense of quadratic equation this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- Quadratic polynomial - A polynomial is an expression; a quadratic equation imposes an equality involving a polynomial of degree two.
- Quadratic function - A quadratic function maps inputs to outputs; a quadratic equation asks which inputs satisfy an equality.
- Linear equation - A linear polynomial equation has degree one rather than two.
- Quadratic inequality - An inequality compares expressions using an order relation rather than equality.
- General second-degree equation in two variables - It can include x², xy, and y² terms and describe a conic; the sense covered here has one variable.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of quadratic equation this model covers, and on what evidence? provenance
What the second pass must settle
- Should this registry entry cover arbitrary coefficient fields and rings, including characteristic two, or establish real and complex arithmetic as its default scope?
- Should equations reducible to a quadratic by substitution be represented here as derived equations or owned entirely by neighbouring equation models?
- What identity rule should distinguish an equation from scalar multiples and other representations with the same admissible solution set?
- What numerical error evidence and tolerance conventions should be required before an approximate solution is marked fully solved?
- Does an existing Vercy world model already own this concept, and how should the supplied INF / INF.MED placement align with its treatment as a mathematical constraint?