Our story begins in 1545. In that year the Italian mathematician Girolamo Cardano published Ars Magna (The Great Art), a 40-chapter masterpiece in which he gave for the first time an algebraic solution to the general cubic equation  

            .

    Cardano did not have at his disposal the power of today's algebraic notation, and he tended to think of cubes or squares as geometric objects rather than algebraic quantities.  Essentially, however, his solution began with the substiution .  This move transforms    into the cubic equation    without a squared term, which is called a depressed cubic and can be written as

            .

You need not worry about the computational details, but the coefficients are    and  .

Exploration.

(c) 2006 John H. Mathews, Russell W. Howell