{
    "name": "Quartic Equation Calculator",
    "url": "https://calculators.geekmidst.com/calculators/quartic-equation-roots-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/quartic-equation-roots-calculator.md",
    "category": "Algebra, geometry & advanced math",
    "description": "Solve a fourth-degree equation with Ferrari’s method: the depressed quartic, its resolvent cubic and all four real or complex roots.",
    "summary": "Enter the five coefficients of ax⁴ + bx³ + cx² + dx + e = 0. Ferrari’s method turns the quartic into a cubic and two quadratics, and the calculator shows each stage.",
    "also_called": [
        "Quartic formula calculator",
        "Fourth-degree equation solver",
        "Biquadratic equation calculator"
    ],
    "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 (x⁴ coefficient)",
            "type": "number",
            "default": "1"
        },
        {
            "name": "b",
            "label": "b (x³ coefficient)",
            "type": "number",
            "default": "-1"
        },
        {
            "name": "c",
            "label": "c (x² coefficient)",
            "type": "number",
            "default": "-19"
        },
        {
            "name": "d",
            "label": "d (x coefficient)",
            "type": "number",
            "default": "49"
        },
        {
            "name": "e",
            "label": "e (constant)",
            "type": "number",
            "default": "-30"
        }
    ],
    "outputs": [
        {
            "key": "roots",
            "label": "Roots",
            "format": "text"
        },
        {
            "key": "x1",
            "label": "x₁",
            "format": "text"
        },
        {
            "key": "x2",
            "label": "x₂",
            "format": "text"
        },
        {
            "key": "x3",
            "label": "x₃",
            "format": "text"
        },
        {
            "key": "x4",
            "label": "x₄",
            "format": "text"
        },
        {
            "key": "nature",
            "label": "Nature of the roots",
            "format": "text"
        },
        {
            "key": "depressed",
            "label": "Depressed quartic",
            "format": "text"
        },
        {
            "key": "resolvent",
            "label": "Resolvent cubic",
            "format": "text"
        },
        {
            "key": "method",
            "label": "Method used",
            "format": "text"
        }
    ],
    "formula": {
        "text": "x = y − b/4a gives y⁴ + py² + qy + r = 0;  resolvent 8m³ + 8pm² + (2p² − 8r)m − q² = 0;  then y² − √(2m)·y + (p/2 + m + q/(2√(2m))) = 0 and y² + √(2m)·y + (p/2 + m − q/(2√(2m))) = 0",
        "variables": {
            "a … e": "the coefficients (a ≠ 0)",
            "p, q, r": "coefficients of the depressed quartic",
            "m": "a positive root of the resolvent cubic"
        }
    },
    "worked_example": {
        "inputs": {
            "a": "1",
            "b": "-1",
            "c": "-19",
            "d": "49",
            "e": "-30"
        },
        "steps": [
            "Substitute x = y + 1/4 to remove the cubic term: y⁴ − (155/8)y² + (315/8)y − 4851/256 = 0.",
            "The resolvent cubic 8m³ − 155m² + (7219/8)m − 99225/64 = 0 has the positive root m = 81/8, so √(2m) = 9/2.",
            "The two quadratics give y = 11/4, 7/4, 3/4 and −21/4; adding 1/4 gives x = 3, 2, 1 and −5."
        ],
        "result": "x⁴ − x³ − 19x² + 49x − 30 = 0 has the roots −5, 1, 2 and 3."
    },
    "interpretation": "A quartic with real coefficients has 4, 2 or 0 real roots, with any non-real roots in conjugate pairs. When q = 0 the depressed quartic is a quadratic in y² (a biquadratic) and the resolvent is not needed.",
    "assumptions": [
        "Coefficients are real; a must not be 0."
    ],
    "limitations": [
        "Roots are shown to 10 significant figures; exact radical forms are not written out."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Is Ferrari’s method accurate?",
            "answer": "Used naively it can lose digits, especially near repeated roots. Here the depressed form is computed exactly, the resolvent root is refined, and every root is polished with Newton’s method on the exact square-free part of the polynomial, so repeated roots come out to full precision."
        },
        {
            "question": "Why is there no general formula for degree five?",
            "answer": "The Abel–Ruffini theorem shows that a general quintic cannot be solved with radicals. Use the polynomial roots calculator for higher degrees."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Cubic Equation Calculator",
            "url": "https://calculators.geekmidst.com/calculators/cubic-equation-roots-calculator"
        },
        {
            "name": "Polynomial Roots Calculator",
            "url": "https://calculators.geekmidst.com/calculators/polynomial-roots-calculator"
        },
        {
            "name": "Quadratic Formula Calculator",
            "url": "https://calculators.geekmidst.com/calculators/quadratic-formula-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}