178 On extracting the Roots of Equations. 



peared ; therefore the initial figure of another real root is 0*4;. 

 One method of discovering impossible roots is as follows; viz. 



When any coefficient of an equation becomes zero, and if 

 the signs of the adjacent coefficients on each side are like, that 

 equation contains at least two impossible roots. 



Another rule which I can demonstrate, is as follows: 



When any column of coefficients has a minimum, and if 

 two changes of signs take place in the transformed equations, 

 at that minimum the original equation has two impossible roots. 

 We shall find the fourth coefficients of the equations ina'— O'l, 

 x — 0-2, and x — 0'3 to be respectively — 8-546, — 8'368, and 

 — 8*4'4'2, which show a minimum and consequently two impos- 

 sible roots. 



Extraction of the Roots of Equations. 



If w^e find the initial figure or figures of a root, we may ex- 

 tract the root of the original equation in x by extracting the 

 root of the transformed equation, thus : Divide the absolute 

 number, annexing a cipher, by the coefficient of the single 

 power, and the first figure of the quotient is the next figure of 

 the root of the original equation to the initial figure or figures 

 already found. 



Proceed in this manner from one denohiination to another, 

 until as many figures are found as may be thought necessary. 



Example. — Find that root of the equation jr* — 5x^-\-3x* — 

 9x + 4 = of which the initial figure has been found to be 

 0*4, the corresponding transformed equation being 



(jc_4)t_34(j;_4)^_204(jr— 4)'^ — 874<4.(a'— 4)4-5856 = 0. 



Now by dividing 58560 by 8744 we obtain '6; and here 

 the whole are regarded as if they had been integers and there- 

 fore the next figure tenths. 



Opei'ation. 

 -8744 



1_34 -204 



— 334 



— 3S8 —22404 

 —322 —24372 



— 316 —26304 

 -3155 



— 3150 -2646175 



— 3145 -2661925 



— 3140 —2677650 

 -31392 



— 31384—268016136 



—3 1 376— 268267208 - 9053340629088 



-31368—268518216-9055486766752+4624531217296. 

 When the entire and correct root of the original equation in ^, 

 as far as the number of figures go, is x=-4658. 



-8878424 

 -9024656 



— 9037886875 

 -9051196500 



+5856 (-658 



+ 5289456 



+ 7705125625 



