eUfilC AND HiaiJER E(^UTI0N'3. 37^ 



equation one degree lovv-er, and thence find a -third root, 

 and so on, till the equation be reduced to' a quadratic j 

 then the two roots of this being found, by the method of 

 completing the square, they will make up the re^nainder 

 of the roots. Thus, in the foregoing equation, having 

 found one root to be 1*02804, connect it by jniuiis with 

 a; for a divisor, and take the given equation with the known 

 term transposed for a dividend : thus, 



^'— i'o28o4).v'~i5jtf* + 65.v— 5c(.x*-i3-97T9dv-|-48'<55€27i:ro. 



Then the two roots of this quadratic equation, or 

 «* — 13*97 196^;=: — 48*63627, by completing the square, 

 are 6*57653 and 7*39543, which are also the other two 

 roots of the given cubic equation. So that all the three 

 roots of that equation, viz. a;^— i5.v*4"^3^— 5^* 



are 1*02804 



and 6*57653 

 and 7*39543 



Sum 15*00000 



And the sum of all the roots is found to be 15, being 

 equal to the coeiTicient of the second term of the equation, 

 which the sum of the roots always ought to be, when they 

 are right. 



Note 3. It is also a particular advantage of the fore- 

 going rule, that it is not necessary to prepare the equa- 

 tion, as for other rules, by reducing it to the usual final 

 form and state of equations. Because the rule may be ap- 

 plied at once to an unreduced equation, though it be ever 

 so much embarrassed by surd and compound xjuantities. 

 As in the following example : 



3. Let it be required to find the root .v of the equation 



^i44A;'—A;^ + 2pf4-^i96;c'—.v' 4-241' =114, or 

 the value of x in it. 



By 



