Substituting, in (3) and (4), the values of x, x', in (1) and (2), we have and or and l.nl.exlog. n, These two latter equations express the constant ratio which the logarithms of n have to each other, in the two systems. 198. Since the only systems in use are the common and Naperian, let a=10, e 2.718281828459... (Art. 190); and let . express the common, and log. the Naperian logarithm: then we have which is the modulus of the common system. The common logarithms may, therefore, be calculated either by substituting this value of M in the general formula of Art. 195, or by multiplying the Naperian logarithm by the modulus. The former method is, of course, by far the more direct and expeditious: in this way we have log. 5·86858896 (+3.9$+5.95 log. 10 log. 2+ log.5 = 1.000000. The logarithms of all numbers are calculated in the same manner; and thus a table of logarithms is formed, for the purpose of simplifying arithmetical calculations (Art. 192). The method of using these is explained in books of tables. CHAP. XII.—THEORY OF QUADRATIC EQUATIONS. ART. 199. DEF. I. Any number which, being substituted for the unknown quantity in an equation, renders its two sides identical, is called a root of the equation. 200. II. A cubic equation, or an equation of the third degree, is one in which the unknown quantity is in the third power. 201. We have seen in Art. 153 that one general forin of a quadratic equation is x2+px=q, or x2+px-q=0. Now, if r be a root of this equation, that is, a value of x, we shall have, by substitution, whence or hence, that is, p2 + pr=q; x2+px=r2+pr, (x+r)(x—r)+p(x—r)=0; (x—r)(x+r+p)=0: from which last equation it follows, that, if r be a root of the equation x2+px=q, the left member is divisible by x-r; and conversely, if x-r divide the left member without remainder, r is a root of the equation. 202. Since a product becomes zero when either of its factors becomes zero, (x—r)(x+r+p)=0 both when x-r0, which gives xr, and when x+r+p=0, which gives x-r―p: hence, r being one root, —r—p is the other; and hence a quadratic has two roots. 203. Since x2+px—q= (x—r)(x+r+p), it follows that a quadratic equation is the product of two factors, which are of the first degree with respect to x. Ал hence, the sum of the roots of a quadratic equation is equal to the coefficient of the first power of the unknown with its sign changed. 205. Since r2+pr-q=0, transposing, and changing all the signs, we have —r2 —pr=—q, or r(—r—p)=—q; and, therefore, the product of the roots of a quadratic equation is equal to the absolute term with its sign changed.* 206. The general value of x (Art. 154), * We might also deduce these properties immediately from the In the four values of the first two of these forms, since the quantities under the radical sign are both positive, the square root can be extracted either exactly or approximately, and therefore the roots of the equation are real; and since √(b2+4 ac) is necessarily a greater number than b, the roots will be positive when the sign of the radical is +, and negative when the sign of the radi cal is Again, in the four values of the last two of these forms, in order to render the evolution possible, and the roots real, we must have b2 >4 ac; but if b2< 4 ac, the roots are both imaginary: and since (b2-4ac) is a less number than b, the roots will be negative when b is positive in the equation, and positive when b is negative.* which is the other value; and therefore, if r is one root, -r-p is the other. Again, transposing the right members of these two values, we have 2+2 −√(222+2) = 0, and 2+2 +√(22+9)=0. Multiply these two together, observing that the one is the difference, and the other the sum of the quantities + and ✓ (22+2), and that, therefore, the product is the difference of the squares: hence we have By reduction this becomes x2+px-q=0, which is the same as is proved in Art. 203. which results agree respectively with what has been shown in Art. 204, 205. * This Art. may be illustrated by examples taken from compound quadratics, p. 222, et seq. |