quadratic equation

Univariate quadratic test

Recognition requires a designated unknown and a polynomial form of degree exactly two in that unknown.

1. Which symbol is the unknown, and which remaining symbols are fixed coefficients or parameters? definition
2. After collecting terms through valid operations, is the highest nonzero power of the unknown exactly two? boundary

Nonzero leading coefficient

The quadratic classification requires a ≠ 0; a = 0 produces a linear or constant equation that must be handled separately.

1. What are a, b and c in ax² + bx + c = 0, and are their values exact, approximate or parameter-dependent? measurement
2. For which parameter values does a vanish, and what equation remains in each such case? boundary

Permitted arithmetic

Division, square roots and factorisation must be interpreted in the declared algebraic setting; the familiar quadratic formula is not universally applicable.

1. In which domain do the coefficients lie, and in which domain must the unknown take values? definition
2. Are division by 2a and the required square-root operations available in this setting? action

Admissible root filter

Roots of the polynomial are candidates until checked against exclusions, interval bounds, integrality requirements and contextual constraints.

1. Which restrictions on the unknown were inherited from denominators, radicals, substitutions or the originating problem? provenance
2. Which polynomial roots satisfy every recorded restriction, including any unit and sign requirements? boundary

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.

1. Which standard, factored or completed-square forms are available in the declared domain? definition
2. Does expansion or another valid identity establish that each proposed form preserves the original equation? action

Solution-set preservation

Multiplication, division, squaring and changes of unknown require explicit checks for lost roots, introduced roots and excluded values.

1. Which steps require a divisor to be nonzero or a substitution to be invertible on the admissible domain? boundary
2. How will candidates be mapped back and checked when the quadratic was obtained from a different original equation? action

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.

1. What is b² − 4ac, and is its sign established exactly or only estimated numerically? measurement
2. Given the coefficient and solution domains, how many distinct roots exist and what are their multiplicities? definition

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.

1. Do zero coefficients, a visible factorisation or a perfect-square form permit a simpler justified solution? action
2. Should the result be exact, parameter-conditional or numerical, and which applicable method supports that requirement? action

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.

1. Does each candidate satisfy the original equation exactly or to a justified numerical tolerance? action
2. What establishes that no admissible root or exceptional parameter case was omitted? boundary

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.

1. What coefficient uncertainty and arithmetic precision affect the discriminant and computed roots? measurement
2. Is rescaling, a cancellation-avoiding evaluation or higher precision needed to justify the requested accuracy? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of quadratic equation this model covers, and on what evidence? provenance

Kinds and varieties

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1. Which of these kinds and varieties hold for the sense of quadratic equation this model covers, and on what evidence? provenance

Real-world use

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1. Which of these real-world use hold for the sense of quadratic equation this model covers, and on what evidence? provenance

Typical measurements

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1. Which of these typical measurements hold for the sense of quadratic equation this model covers, and on what evidence? provenance

Failure modes and hazards

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1. 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.

1. 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