Quadratic Equation Solver & Discriminant
Solve quadratic equations with exact fractional or radical roots, a step-by-step derivation, the vertex and a parabola graph.
About this quadratic equation solver & discriminant
Solve quadratic equations with exact fractional or radical roots, a step-by-step derivation, the vertex and a parabola graph Computes exact algebraic and numerical metrics directly in your browser using IEEE 754 precision without round-trip latency.
Formula & calculation mechanics
The execution engine processes input tokens and applies transformations in the browser sandbox. Specifically, For ax²+bx+c=0 with a≠0, Δ=b²−4ac. Positive Δ gives two real roots, zero gives a repeated root, and negative Δ gives a complex conjugate pair (i²=−1). The vertex is (−b/(2a), −Δ/(4a)).
Worked numerical calculation
Scenario: What does the example produce?
Sample input:
Processing: Enter the three rational coefficients, with a nonzero. Read the discriminant, exact roots and vertex. Follow the algebraic steps and inspect the graph; approximations are separate.
Illustrative output:
Domain range & precision boundaries
Exact rational input: at most 64 UTF-16 code units per field, 30 mantissa digits (including leading zeros) or 30 digits per fraction integer; decimal exponent −30…30. Fractions contain integers only. a must be nonzero. These are browser workload limits, not usage quotas. Decimal approximations use binary64 and may merge nearby roots; exact expressions remain authoritative. Fractions are reduced and denominators made positive. Square factors 2² through 1000² and a remaining perfect square are extracted exactly. Larger residual radicals may still contain square factors; they remain exact, but are not guaranteed to be fully simplified. The steps clear denominators and remove a common factor to obtain A, B, C; D is the discriminant of that equivalent equation. The blue curve shows a finite segment of the real function; red marks the vertex and green marks real roots. Axes at zero appear only when inside the displayed range. An exact quadratic Bézier shape is mapped to approximate screen coordinates; x and y have separate scales. Complex roots are not points on this real graph. Full exact axis ranges are printed below. Runs locally in your browser, with no added calculation dependency, fee or external service. Available offline after loading.
Calculations in Quadratic Equation Solver & Discriminant use standard IEEE 754 precision or exact rational models. Verify significant accounting or tax figures with authoritative references.
In-browser calculation boundary
Formula calculations and numeric evaluations in Quadratic Equation Solver & Discriminant run synchronously in the browser JavaScript engine with zero server telemetry.
Mathematical formulas & precision standards
Mathematical constants, IEEE 754 floating-point standards, and numerical analysis specifications:
- NIST Digital Library of Mathematical Functions, §1.2 — Elementary Algebra
- ECMAScript 2026, §6.1.6.1 Number Type
Quadratic Equation Solver & Discriminant verification · Audit standards
How to use it
- Enter, paste, or select your input data into the Quadratic Equation Solver & Discriminant workspace controls.
- Review parameter fields, units, or options configured for Quadratic Equation Solver & Discriminant.
- Click the action button or observe immediate live calculations rendered by Quadratic Equation Solver & Discriminant.
Frequently asked questions
What does the discriminant tell me?+
For ax²+bx+c=0 with a≠0, Δ=b²−4ac.
How are exact expressions simplified?+
Boundary constraints concerning "How are exact expressions simplified?" prevent invalid parameter state mutations client-side.
What input and graph limits apply?+
Runtime handling of "What input and graph limits apply?" executes synchronously in your browser tab without remote delays.
What does the example produce?+
Security isolation for "What does the example produce?" ensures zero transmission beyond your local workstation.