When 9ac, the expression + /sr + c has two real and differing roots, contained in the formula' _ 6 + 4ac) 2a and lea always the same sign as a, except when x lies between those roots. Every change of signs in passing from a to L and from b to c indicatea a positive root, and every continuation a negative root : and when one root is positive and one root negative, the positive or negative root is numerically the greater, according as (a, 6) shows a change or continuation. When x= : 2a, the expression is at its numerical maximum between the two roots, its then value being : 4a.
When 4ac, the expression bx + c is a perfect square wait respect to x, and absolutely no if a be a square. The two roots become equal, and each equal to 6 : 2a. The expression now never differs in sign from a.
When 6'<9ae, the two roots become imaginary, the expression always has the sign of a, and is numerically least when x= : being then (4ac : 9a.
3. The equation of the third degree (or cubic) has been separately considered in the article IRREDVCIBLE CASE.
4. Nothing belongs particularly to the equation of the fourth degree (or biquulratie) except the recital of the various modes in which the solution is reduced to that of a cubic. Tho various modes aro (Ea tinguiehed by the names of their inventors.
Ferrari. Let + + b + c = 0. This can be transformed into + = ( a) Lx + c: make the second side a perfect square; that is, find the value of r from L'=4 e) a), or See + 4ac = 0; the extraction of the square root then reduces the hicputdratie to a couple of quadratics.
Des Cartes. Let xl + +Ls c =(x" + Jp x + f) (al px +g), which gives 9 + -r= (.1 -.0 = = e or + 2ap* + (a' 4c) p b' = 0 : find a positive root of this equation (it certainly has one), and from it find g and f ; then the roots of a/p. f 0, and s/p.x+ y = 0, are those of the given equation.
Thomas Simpson gave a modification of Ferrari's method, and Euler one of that of Des Cartes. (Murphy's Theory of Equations (' Library of Useful Knowledge'), pp. 54, 55.) The theory of equations of all degrees is to be divided into two dis tinct parts; the numerical solution, and the general properties of the roots and the expression(' themselves. The numerical solution must be carefully distinguished from the general solution ; ,tho former term applying to any mode of approximating to a single root, tho latter to any mode of exhibiting a general expression for the roots. We shall
begin by the general properties of the roots : the expression in ques tion being rex, or a + a,a^ + + + + 1. If r be a root of cp.r, or if Or= 0, then stz is dis-isigle by xr, and the quotient is smoother such expression of the (n Dth degree, every root of which is also a root of rtsrs and every number which is not a root (t excepted) is not a root of fir. Hence rar cannot have more roots than it has dimensions, or cannot have more than is roots.
2. When the expression rex is divisible by (xr)s, it is said to have m roots each equal to r; and when this its the case, the substitu tion of r + y for a would give an expression in which y" is the lowest power of y.
3. Every expression has as many roots as it has dimensions. This proposition is of complex proof, but it begins to occupy its proper place in elementary works, especially on the continent.
4. We may now refer to STURM'S THEOREM, to FOHT/SY'S theorem (given in the article just cited), to Des Cartes' theorem, a very limited particular case of Fourier's, and to Homer's adaptation of, and addition to, the old method of numerical solution by Vieta (an account of the history of this last problem is given in the Companion to the Almanac' for 1839). [INvoLuTioN aND EVOLUTION.] We have then, since the beginning of this century, a complete theoretical mode of deter mining the number of roots, real or imaginary, between any given limits; both exceedingly difficult in the complication of the opera tions which they require. Also, a mode of easy application, though not theoretically perfect, of determining the limits between which the real roots lie ; and a process for the numerical solution which places that question upon the same footing as the common extrac tion of square, cube, &c., roots; making those extractions them selves, except only in the case of the square root, much easier than before. In Cauchy's theorem, now beginning to be generally known, and which was given in the Penny Cyclopedia,' we have a theoretical mode of determining the imaginary roots. And Homer's method has been extended to their computation. But it seldom happens that the actual determination of imaginary roots is required.