Elements of Algebra

508 ADDITIONS. CHAP. II. have $-\frac{B}{2A}$ for the whole real part of these roots, which must consequently be taken for $a$. So that we shall only have to reduce the fraction $-\frac{B}{2A}$, abstracting from the sign it may have, to a continued fraction, by the method of Art. 4, and then deduce from it the series of converging fractions (Art. 10), which will necessarily terminate. This being done, we shall successively try for $p$ the numerators of these fractions, and the corresponding denominators for $q$, taking care to give $p$ and $q$ the same, or different signs, according as $-\frac{B}{2A}$ is a positive, or negative number. In this manner, we shall find the values of $p$ and $q$, that may render the formula proposed a minimum. *Example*. Let there be proposed, for example, the quantity $49p^2 - 238pq + 290q^2$. Here, we shall have $A = 49$, $B = -238$, $c = 290$; wherefore $B^2 - 4AC = -196$, and $-\frac{B}{2A} = \frac{238}{98} = \frac{17}{7}$. Working with this fraction according to the method of Art. 4, we shall find the quotients $2, 2, 3$; by means of which, we shall form these fractions (see Art. 20), $2, 2, 3.$ $\frac{1}{0}, \frac{2}{1}, \frac{5}{2}, \frac{17}{7}$. So that the numbers to try with will be $1, 2, 5, 17$, for $p$, and $0, 1, 2, 7$, for $q$. Now, denoting the quantity proposed by $P$, we shall have $p$ $q$ $P$ $1$ $0$ $49$ $2$ $1$ $10$ $5$ $2$ $5$ $17$ $7$ $49;$ whence we perceive, that the least value of $P$ is $5$, which results from these suppositions $p = 5$, and $q = 2$; so that we may conclude, in general, that the given formula can never become less than $5$, while $p$ and $q$ are whole numbers; so that the minimum will take place, when $p = 5$, and $q = 2$. *Second case*, when $B^2 - 4AC > 0$. 33. As, in the present case, the equation $Ax^2 + Bx + c = 0$,