{
  "slug": "quadratic-equation",
  "title": "Quadratic Equation Solver",
  "heading": "Quadratic Equation Solver",
  "category": "math",
  "url": "https://www.revenuelab.fyi/toolbox/quadratic-equation",
  "summary": "Roots, discriminant, and vertex of ax² + bx + c.",
  "description": "Solves any quadratic using the quadratic formula. The discriminant tells you the shape of the answer before you compute it: positive gives two real roots, zero gives one, negative gives a complex pair.",
  "formula": "x = (−b ± √(b² − 4ac)) ÷ 2a.",
  "dateModified": "2026-09-30",
  "run_url": "https://www.revenuelab.fyi/api/public/calc?tool=quadratic-equation",
  "inputs": [
    {
      "id": "a",
      "label": "a (x² coefficient)",
      "kind": "number",
      "hint": null,
      "default": 1,
      "unit": null,
      "min": null,
      "max": null
    },
    {
      "id": "b",
      "label": "b (x coefficient)",
      "kind": "number",
      "hint": null,
      "default": -5,
      "unit": null,
      "min": null,
      "max": null
    },
    {
      "id": "c",
      "label": "c (constant)",
      "kind": "number",
      "hint": null,
      "default": 6,
      "unit": null,
      "min": null,
      "max": null
    }
  ],
  "outputs": [
    {
      "id": "roots",
      "label": "Roots",
      "format": "raw",
      "hint": null,
      "primary": true
    },
    {
      "id": "nature",
      "label": "Root type",
      "format": "raw",
      "hint": null,
      "primary": false
    },
    {
      "id": "discriminant",
      "label": "Discriminant (b² − 4ac)",
      "format": "decimal",
      "hint": null,
      "primary": false
    },
    {
      "id": "vertexX",
      "label": "Vertex x",
      "format": "decimal",
      "hint": null,
      "primary": false
    },
    {
      "id": "vertexY",
      "label": "Vertex y",
      "format": "decimal",
      "hint": null,
      "primary": false
    }
  ],
  "worked_example": {
    "inputs": [
      "a (x² coefficient): 1",
      "b (x coefficient): -5",
      "c (constant): 6"
    ],
    "outputs": [
      "Roots: x = 3.0000 or 2.0000",
      "Root type: Two distinct real roots",
      "Discriminant (b² − 4ac): 1",
      "Vertex x: 2.5",
      "Vertex y: -0.25"
    ]
  },
  "how_to": {
    "title": "How to use this",
    "steps": [
      "Enter a (x² coefficient).",
      "Enter b (x coefficient).",
      "Enter c (constant).",
      "Read your roots on the right — it updates as you type.",
      "Hit Share to keep the scenario or send it to someone."
    ]
  },
  "scenarios": [
    {
      "name": "Conservative",
      "description": "Lower-end numbers — what if things land soft?",
      "values": {
        "a": 0.6,
        "b": -3,
        "c": 3.5999999999999996
      }
    },
    {
      "name": "Typical",
      "description": "Defaults — the most common real-world setup.",
      "values": {
        "a": 1,
        "b": -5,
        "c": 6
      }
    },
    {
      "name": "Ambitious",
      "description": "Higher-end numbers — what if things really pop?",
      "values": {
        "a": 1.6,
        "b": -8,
        "c": 9.600000000000001
      }
    }
  ],
  "limitations": [
    "Results are rounded for display; the underlying calculation keeps full precision.",
    "Very large or very small inputs may hit floating-point limits in the browser.",
    "Inputs outside the accepted range are clamped rather than rejected."
  ],
  "faq": [
    {
      "q": "How does the quadratic equation solver work?",
      "a": "Solves any quadratic using the quadratic formula. The discriminant tells you the shape of the answer before you compute it: positive gives two real roots, zero gives one, negative gives a complex pair. The underlying maths is: x = (−b ± √(b² − 4ac)) ÷ 2a."
    },
    {
      "q": "What do I need to enter?",
      "a": "3 values: a (x² coefficient), b (x coefficient), and c (constant). Each field starts with a sensible default, so you can change one number at a time and watch the result move."
    },
    {
      "q": "What does the roots result mean?",
      "a": "It is returned as a plain value and updates live as you edit the inputs, so you can compare two or three versions of a scenario in a few seconds."
    },
    {
      "q": "Is this calculator free, and do I need an account?",
      "a": "Yes, it's free, and no account is required. Nothing you type is stored on our servers — the maths runs entirely in your browser."
    },
    {
      "q": "How accurate is the result?",
      "a": "It applies the standard formula exactly, so the arithmetic is precise. Results are rounded for display; the underlying calculation keeps full precision."
    },
    {
      "q": "Who is this tool for?",
      "a": "It's built for students, developers, and anyone who needs the answer right now without setting up a spreadsheet."
    }
  ],
  "related": [
    "https://www.revenuelab.fyi/toolbox/slope",
    "https://www.revenuelab.fyi/toolbox/pythagorean"
  ],
  "license": "CC-BY-4.0",
  "citation": "RevenueLab — Quadratic Equation Solver (https://www.revenuelab.fyi/toolbox/quadratic-equation)"
}