Quartic Formula Calculator

A quartic equation is a polynomial equation of degree 4, ax^4 + bx^3 + cx^2 + dx + e = 0. It has exactly four roots once complex roots and repeats are counted, and because the coefficients are real, any complex roots come in conjugate pairs. So a quartic has four real roots, two real roots and one complex pair, or two complex pairs.

Step 1: rational roots. Before any formula, the rational root theorem lists every fraction p/q that could be a root: p divides the constant term and q divides the leading coefficient. For x^4 - x^3 - 2x^2 - 3x - 1 the candidates are just 1 and -1, and neither works, so this quartic has no rational root. When one does work, it is divided out and what is left is a cubic.

Step 2: the depressed quartic. Dividing by a and substituting x = y - b/(4a) removes the cubic term. Here b/(4a) is -1/4, so x = y + 1/4, and the equation becomes y^4 - (19/8)y^2 - (33/8)y - 483/256 = 0, which has the form y^4 + Py^2 + Qy + R = 0.

Step 3: Ferrari's resolvent cubic. Ferrari's idea was to add a term m to both sides so the left becomes a perfect square, (y^2 + P/2 + m)^2, and choose m so the right is a perfect square too. That happens when m solves the resolvent cubic 8m^3 + 8Pm^2 + (2P^2 - 8R)m - Q^2 = 0, here 8m^3 - 19m^2 + (211/8)m - 1089/64 = 0. It has the root m = 9/8, and 2m = 9/4 is the square of 3/2.

Step 4: two quadratics. With that m the depressed quartic is a difference of two squares, (y^2 - 1/16)^2 - ((3/2)y + 11/8)^2, which factors into two quadratics. Shifting back to x gives x^4 - x^3 - 2x^2 - 3x - 1 = (x^2 - 2x - 1)(x^2 + x + 1). The calculator multiplies the two factors back together in exact fractions to confirm the split.

Step 5: the quadratic formula on each. x^2 - 2x - 1 = 0 gives x = 1 - √2 and x = 1 + √2, about -0.4142135624 and 2.414213562. x^2 + x + 1 = 0 has discriminant -3, so it gives the complex pair x = -1/2 ± (√3/2)i.

When the quartic does not split so neatly. The resolvent always has a root, but often not a rational one, and then the quartic formula writes each root as nested cube roots and square roots that run across several lines. Evaluated in decimals, that formula also loses digits to cancellation. So when a quartic does not factor over the rationals, the calculator finds the four roots numerically with the Aberth-Ehrlich method and then checks each one. A real root is checked by evaluating the polynomial in exact arithmetic just below and just above it: a sign change proves a root lies between. A complex root is checked by substituting it back. A quartic with no x^3 or x term, such as x^4 - 10x^2 + 1 = 0, is a quadratic in x^2 and comes out exactly as x = ±√(5 ± 2√6).

Why stop at four? Formulas in radicals exist for degrees 2, 3 and 4. Abel and Galois showed that no such formula exists for degree 5 or higher, so past quartics the roots are found numerically. This calculator solves those too, with the same checks.