How to use this calculator
- Enter or choose a (x⁴ coefficient).
- Enter or choose b (x³ coefficient).
- Enter or choose c (x² coefficient).
- Enter or choose d (x coefficient).
- Enter or choose e (constant).
- The roots updates as you type. Use Copy result or Share to keep it.
Method
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
- a … e
- the coefficients (a ≠ 0)
- p, q, r
- coefficients of the depressed quartic
- m
- a positive root of the resolvent cubic
Worked example
- 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.
x⁴ − x³ − 19x² + 49x − 30 = 0 has the roots −5, 1, 2 and 3.
What the result means
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 and limits
- Coefficients are real; a must not be 0.
- Roots are shown to 10 significant figures; exact radical forms are not written out.
Questions
Is Ferrari’s method accurate?
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.
Why is there no general formula for degree five?
The Abel–Ruffini theorem shows that a general quintic cannot be solved with radicals. Use the polynomial roots calculator for higher degrees.