The rules for the solution of the simpler forms of E. are to be found in all elementary text-books of algebra. It must suffice to notice here a few of the leading general properties of equations. By the roots of an equation are meant those values real or imaginary of the unknown which satisfy the equality; and it is a property of every equation to have as many roots and no more as there are units in its degree. Thus, a quadratic equation has two roots; a cubic equation, three ; and a biquadratic, four. The quadratic equation + 5x — 36 = 0 has two roots, 9 and — 4, which will be found to satisfy it. Further, the expression x' + 5x — 36 = — 9) (z + 4) = 0; and generally if the roots of an equation F(x) = a'n „ .
±A,x±Ao = 0
(x and = F(x) = 0.
Hence, and from observing the way in which, in the multiplication of these factors, the co-efficients A,, Ao are formed, we arrive at the following important results: = the sum of the roots, with their signs changed.
= the sum of the products of every two roots, with their signs changed. _3 = the sum of the products of every three roots, with their signs changed. = the product of the roots, with their signs changed.
The factors, it will be observed, arc formed thus: If + a, be a root, then x = a,, and x — a, = 0 is the factor. If the root were — a,, then z = — a, ; and the factor would be x-{- a, = 0.• Observing now the way in which, in multiplying a series of such factors, the co-ancients of the resulting polynominal are formed, we arrive at this: that a complete equation cannot have a greater number of positive roots than these changes of sign from to — and from — to in the series of terms forming its first member; and that it cannot have a greater number of negative roots than there are permanencies or repetitions of the same sign in proceeding from term to term. From the same source, many other general properties of E., of value in their arithmetical solution, may be inferred. The subject is, however, to vast to be more than glanced at here.