Elements of Algebra

CHAP. XVI. OF ALGEBRA. 297 that $1 = \frac{1}{x-1}$; multiplying therefore by $x-1$, we have $x-1=1$, and $x=2$. 801. Beside these methods of determining the roots of an equation by approximation, some others have been invented, but they are all either too tedious, or not sufficiently general*. The method which deserves the preference to all others, is * This remark does not apply to the method of finding the roots of equations of all degrees, and however affected, by The Rule of Double Position. In order, therefore, that this chapter might be more complete, we shall explain this method as briefly as possible. Substitute in the given equation two numbers, as near the true root as possible, and observe the separate results. Then, as the difference of these results is to the difference of the two numbers; so is the difference between the true result, and either of the former, to the respective correction of each. This being added to the number, when too small, or subtracted from it, when too great, will give the true root nearly. The number thus found, with any other that may be supposed to approach still nearer to the true root, may be assumed for another operation, which may be repeated, till the root shall be determined to any degree of exactness that may be required. *Example*. Given $x^3 + x^2 + x = 100$. Having ascertained by a few trials, or by inspecting a Table of roots and powers, that $x$ is more than 4, and less than 5, let us substitute these two numbers in the given equation, and calculate the results. By the first supposition $\left\{ \begin{array}{l} x=4 \\ x^2=16 \\ x^3=64 \end{array} \right.$ By the second supposition $\left\{ \begin{array}{l} x=5 \\ x^2=25 \\ x^3=125 \end{array} \right.$ 84 . . . Results . . . . . . 155 155 5 100 true result. 84 4 84 Differences 71 1 16 Then, As $71 : 1 :: 16 : \cdot 2253 +$ Therefore $4 + \cdot 2253$, or $4\cdot 2253$ approximates nearly to the true root. If now $4\cdot 2$, and $4\cdot 3$, be taken as the assumed numbers, and substituted in the given equation, we shall obtain the value of $x = 4\cdot 264$ very nearly, the error being only $\cdot 027552256$.