{
    "name": "Quadratic Formula Calculator",
    "url": "https://calculators.geekmidst.com/calculators/quadratic-formula-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/quadratic-formula-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Solve any quadratic equation with the quadratic formula: real or complex roots, the discriminant and the vertex of the parabola.",
    "summary": "Enter a, b and c from ax² + bx + c = 0. The discriminant decides whether there are two real roots, one repeated root or two complex roots.",
    "also_called": [
        "Quadratic equation solver",
        "Quadratic solver"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "a",
            "label": "a (coefficient of x²)",
            "type": "number",
            "default": "1"
        },
        {
            "name": "b",
            "label": "b (coefficient of x)",
            "type": "number",
            "default": "-5"
        },
        {
            "name": "c",
            "label": "c (constant)",
            "type": "number",
            "default": "6"
        }
    ],
    "outputs": [
        {
            "key": "roots",
            "label": "Solutions",
            "format": "text"
        },
        {
            "key": "nature",
            "label": "Type of roots",
            "format": "text"
        },
        {
            "key": "discriminant",
            "label": "Discriminant (b² − 4ac)",
            "format": "number"
        },
        {
            "key": "x1",
            "label": "x₁",
            "format": "text"
        },
        {
            "key": "x2",
            "label": "x₂",
            "format": "text"
        },
        {
            "key": "vertex",
            "label": "Vertex of y = ax² + bx + c",
            "format": "text"
        }
    ],
    "formula": {
        "text": "x = (−b ± √(b² − 4ac)) ÷ 2a;  Discriminant D = b² − 4ac;  Vertex = (−b ÷ 2a, c − b² ÷ 4a)",
        "variables": {
            "a, b, c": "the coefficients, with a ≠ 0",
            "D": "discriminant: > 0 two real roots, = 0 one repeated root, < 0 two complex roots"
        }
    },
    "worked_example": {
        "inputs": {
            "a": "1",
            "b": "-5",
            "c": "6"
        },
        "steps": [
            "D = (−5)² − 4 × 1 × 6 = 25 − 24 = 1.",
            "x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, so x = 3 or x = 2.",
            "Check: (x − 3)(x − 2) = x² − 5x + 6."
        ],
        "result": "x² − 5x + 6 = 0 has two real solutions, x = 3 and x = 2."
    },
    "interpretation": "The real roots are where the parabola y = ax² + bx + c crosses the x-axis. Complex roots mean it never crosses; the vertex is then the closest it gets.",
    "assumptions": [
        "Coefficients are real numbers and a is not 0."
    ],
    "limitations": [
        "Roots are shown to 10 significant digits. The calculator uses the numerically stable form of the formula, so a tiny root next to a huge one is still accurate."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "What if a is 0?",
            "answer": "Then there is no x² term and the equation is linear: bx + c = 0 gives x = −c ÷ b."
        },
        {
            "question": "What does i mean in the answer?",
            "answer": "i is the imaginary unit, √−1. A negative discriminant gives two complex roots p + qi and p − qi, which are conjugates of each other."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Slope Calculator",
            "url": "https://calculators.geekmidst.com/calculators/slope-calculator"
        },
        {
            "name": "Exponent Calculator",
            "url": "https://calculators.geekmidst.com/calculators/exponent-calculator"
        },
        {
            "name": "Triangle Calculator",
            "url": "https://calculators.geekmidst.com/calculators/triangle-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}