ROOT. 



In the fecond cafe, put — (V = c0 ^ z > tncn nas r 

 the three following values : 



/a a 



. x = — 2 \ / — x cof. — . 



V 3 3 



V /-- x cof. ( 6o° + --Y 



V 3 V 3 / 



2. * = 



8/' -4*/' + (2ac- »d)p- a>J+ 4 bd-c' = o 

 ? = v/ (i«* + 2/- *) 



at — c 

 r ~ * ; 



then will the four roots of the propofcd equation be con- 

 tained in the following formulae ; 



The two latter cafes of forms 3 and 4 belong to the ir- 

 reducible cafe, each of which gives three real roots or 

 values of .\- ; whereas the other forms have each only one 

 real root. See IRREDUCIBLE Cafe. 



By infinite Series. 



Let x' + ax = b reprefent any cubic equation, in which 

 a and b may be each either pofitive or negative. 



2 b , 27 l' 



± ,{(fc=*> + ,-,l 



+ */ 



KW— >} 



Allume — — " " -r-=r =■ C, and 



^[2(27^ + 42)] 27J- 



4"' 



The above rule was only generalized by Simpfon, it is 

 originally due to Ferrari, though commonly afcribed to 

 Bombelli. 



By Defcartes' Rule. 



IX. Let x s + a x 7 + b x + c = o, be any biquadratic 

 equation, wanting its fecond term. 



Find the value of y in the double cubic equation 



- ; then, 



x = J> X [I + 



2.5 



■•<{ 



6.9 



+ &c], or 



+ 



.9.12.15 



+ 



fx[i+ I— *• A + 



L 6.9 12 . 15 



18.21 



*C + &c] 



In which laft form, A, B, C, &c. reprefent the pre- 

 ceding terms. 



27 b' + 4<i' 



/ + 2 a y 1 4 ((j 1 — 4 r ) / - b 1 — o ; 



2 . C.. . 17 tnen wl " tne ^ our va luss of a; be comprifed in the formulae 

 6. 9... 21 • i x 



ir li + 



Again, affume 2 • / - = p, 



then, 



f x = ? X [ I - 



and 



27 A' 



3-6 



7 1 - &c], or 



x = a x 



[> 



^ A 



* B 



3.6 9. 12 



•C - &c] 



In which, alfo, A, B, C, &c. are the preceding terms. 



Both thefe feries arc correft analytical expreffions for the 

 value of x, in the general cubic equation .v 4 a x = b ; 

 but they are not equally commodious for the purpofes of 

 folution. 



The former mull be ufed in all cafes when <i is pofitive, 

 as alio when a is negative, and 4^ greater than 54^; 

 and the latter when a is negative, and 4 a lefs than 54 i 1 ; 

 becaufe then ~, in both cafes, will be lefs than unity, and 

 the feries will, therefore, be converging ones. 



Biquadratic Equations. 



By Simpfon' J Rule. 



VIII. Let x* + a x 1 + bx + c x + d = o, be any 

 equation of the fourth d?gree, in which a, b, c, d, may be 

 any numbers at pleafure, pofitive, negative, or zero 



By Euler's Rule. 



X. Let .*:' — ax" — bx — c = o, be any biquadratic equa- 

 tion, wanting its fecond term, a, b, and e, being any num- 

 bers, pofitive, negative, or zero. 



A flume /= \a ; h = — ; g = -rV «' + \ t. 



04 



Then find the three roots of the cubic equation 



V + // + g y - h = O, 



which let be p, 7, and r. 



Then will the four values of .v in the original equation be 

 15. 18 exprefled as follows. 

 When // is pofitive, 



2.5... 14 

 3.6... 18 



1 ! 



'4 



Equations in-gcneral. 



/?y Approximation. — F/V/? Method. 



XI. Let *" + a *" ' + Z x" -' + r *"-• + ,7 a-"- * + &c. 

 O, be any general equation, in which a, b, c, d, Sic. arc 



Find the valui s of p, q, and r, by means of the three any numbers, pofitive, negative, or zero ; then r being an 

 equations approximate value of v, we have 



