CUBIC EQUATION. A rational intogral equation of the third degree is called a cubic equation. It is called binary, ternary, or qua ternary according as it is homogeneous of the third degree in two, three, or four unknowns. The general form of a cubic equation of one un known is ax' bx' cx d = 0. It is shown in algebra that this equation can be reduced to one of the form x' + px + q = 0. Every cubic equation of this form has three roots, of which one is real and the others real or imaginary. The roots will all be real when p is negative, and !*, This is known as the irreducible ease in solving the equation. Only one root is real when p is positive, or when it is negative and < negative ' 2 27 If p is neative and 27 — p (1 two of the roots are equal. The cubic equation may be solved by the following formula, due to Tartaglia and Ferro, Italian mathematicians of the six teenth century, but known as Cardan's formula: _ V\ ± ± Besides Ferro, Tartaglia. and Cardan, Vista,
Euler. and others contributed to the early the ory of cubic equations. In case the roots of a cubic equation are all real their values are more readily calculated by means of trigonometric formulas—e.g. assume x= a cosa, and the equa tion + p,j+ q = 0 may be expressed by eosa + ;-; eosa + = 0. But from trigonometry coiu2a — 3 cosa — = 0; therefore, equat ing corresponding coefficients of eosa, and solv i 4p tug the equations, a = and cos3a = — 4q Hence x may now be computed from 4 27r x 77 • COS a n • cos + a ; n. For history and methods, consult Alatt-hiesse-n. Grand:nig(' der antiken and modernen Algebra der litteralen Gleiehungen (Leipzig, 1896). See also CARDAN, •EROME.