How the solver reaches and checks an answer

Purpose and scope of this methodology

This page explains the method used by the shared-hosting calculator available on this website. It is intended to help learners, teachers, reviewers, and maintainers understand what happens between an entered problem and the displayed result. It does not claim that the engine covers all of algebra.

The current engine is designed for supported one-variable equations and inequalities, common polynomial operations, exact numeric and fraction arithmetic, positive-integer LCM calculations, ratio simplification, polynomial evaluation at an assigned x-value, and polynomial graph preparation. A problem outside that scope should stop with a capability message rather than be forced through an unsuitable rule.

1. Parse and classify the input

The input is normalized into a mathematical structure before solving begins. The parser recognizes supported numbers, the variable x, arithmetic operators, grouping symbols, powers, equality or inequality relations, and linear absolute-value notation. It separates the left and right sides of a statement and constructs polynomial coefficients where the structure allows it.

Classification determines the next method. An equation is not treated as an expression, and factoring is not treated as solving. The same visible formula can therefore produce a different outcome depending on the selected operation and whether a relation symbol is present.

• Equation families: one-variable linear, quadratic, and supported linear absolute-value equations.
• Inequality family: supported one-variable linear inequalities.
• Expression operations: simplify, combine like terms, factor over supported rational cases, and expand polynomial products.
• Numeric operations: exact integer and fraction calculations, paired LCM and GCD results for two to eight positive integers, and two- to six-part ratio simplification with exact teaching steps.
• Function operations: exact polynomial evaluation at x and graph data for polynomial functions through degree four.

2. Preserve exact arithmetic

Integers, terminating decimals, and written fractions are normalized as exact rational values where the selected method supports them. This avoids introducing a rounded decimal merely because an intermediate division is not an integer. For example, 7/3 remains 7/3 rather than being shortened to 2.33.

Quadratic roots may remain in radical form when a square root is irrational. When the complex domain is selected and the discriminant is negative, the result can include i. Fraction views label whether a decimal terminates exactly; repeating decimal displays are secondary approximations while the reduced fraction remains authoritative.

3. Apply named, solution-preserving rules

Each displayed step is tied to an algebra rule such as distribution, combining like terms, adding the same quantity to both sides, dividing by a nonzero coefficient, reversing an inequality after division by a negative value, factoring, or splitting an isolated absolute-value equation into cases.

The working is educational rather than a raw trace of internal code. Consecutive machine operations may be grouped into one meaningful learner-facing step, but intermediate lines are included when they explain an important sign, coefficient, fraction, or branch decision.

1. Show the original or normalized problem.
2. Simplify each side without changing its value.
3. Apply the method required by the classified problem family.
4. State the complete result in the selected number domain.
5. Run the available independent verification and label its strength.

4. Verify candidates and equivalent forms

Equation candidates are substituted into the original statement when exact evaluation is available. The calculator compares the resulting left and right values. For a quadratic, the method must also account for both branches or use a theorem that establishes completeness. Inequalities require boundary and order reasoning, not only substitution of one number.

Expression operations use an equivalence check. The engine compares normalized polynomial coefficients or expands a factored product back to the original expression. This evidence checks the final form independently from the way it was displayed.

Testing, review, and change control

The release checks include representative solver fixtures, PHP shared-hosting tests, JavaScript syntax checks, route checks, metadata checks, internal-link validation, and browser inspection of representative desktop and mobile states. These checks reduce risk but do not prove the absence of every possible defect.

Material changes to solver behavior should be tested against known examples and recorded in the changelog. Content describing capabilities must be updated when the engine changes so the interface, Accuracy page, and actual behavior do not drift apart.