Quartic Equations
The solution of quartic equations was found soon after that of cubic equations. It is due to Ludovico Ferrari (1522–1565), a pupil of Cardano, and it first appeared in Cardano’s “Ars Magna.”
Let
x4 + ax3 + bx2 + cx + d = 0
be an arbitrary quartic equation. By the change of variable x = y −a/4 the cubic term cancels out, and the equation becomes
y4 + py2 + qy + r = 0 (5.8.6)
with
p = b −6(a/4)2
q = c − ba/2 + (a/2)2
r = d − ac/4 + (a/4)2b − 3(a/4)4
Moving the linear terms of (5.8.6) to the right-hand side and completing the square on the left-hand side, we obtain
(y2 + p/2)2 = −qy − r + (p/2)2
If we add a quantity u to the expression squared in the left-hand side, we get
(y2 + p/2 + u)2 = −qy −r + (p/2)2 + 2uy2 + pu + u2 (5.8.7)
The idea is to determine u in such a way that the right-hand side also is easily seen becomes a square. Looking at the terms in y2 and in y, it is easily seen that if the right-hand side is a square, then it is the square of √(2u)y − q/2√q 2u; therefore, we have
−qy −r + (p/2)2 + 2uy2 + pu + u2 = [√(2u)y − q/2√(2u)]2 (5.8.8)
Equating the constant terms, we see that this equation holds if and only if
−r + (p/2)2 + pu + u2 = q2/8u
or equivalently, rearranging terms
8u3 + 8pu2 + (2p2 −8r)u −q2 = 0 (5.8.9)
So, by solving this cubic equation, we can find a quantity u for which equation (5.8.8) holds. Returning to equation (5.8.7), we then have
(y2 + p/2 + u)2 = [(2u)1/2 − q/(2(2u)1/2]2.
hence
y2 + p/2 + u = ± [(2u)1/2 − q/(2(2u)1/2]2.
To complete the discussion, it remains to consider the case where u = 0 is a root of equation (5.8.9), since the calculations above implicitly assume u ≠ 0. But this case occurs only if q = 0 and then the initial equation (5.8.6) is
y4 + py2 + r = 0
which can be easily solved, since it is a quadratic equation in y2.
To sum up solutions of x4 + ax3 + bx2 + cx + d = 0 are if q ≠ 0
x = ε (u/2)1/2 + ε']−u/2 − p/2 − εq/(2(2u)1/21/2 − a/4
where ε and ε' can be independently ±1. If q = 0, the solutions are
x = ε [−p/2 − ε'(p/2))2 − r]1/2 − a/4
Having solved the cubic and quartic by radicals, mathematicians turned to finding a solution by radicals of the quintic
Paolo Ruffini and Niels-Henrik Abel prove (in 1799 and 1829, respectevely) the unsolvability by radicals "of the general equation" of degree n for every n > 4. Although the general equation is unsolvable by radicals, some specific equations of this form are solvable; for example xn − 1 = 0 is solvable by radicals for every n > 4. Galois characterized those equations that are solvable by radicals in terms of group theory. We'll cover this later.