Of CUBIC EQUATIONS. 113 



Applying this rule to our example, we have : 

 for the two equal roots, -"^ = f — -^ = 5 



,. , , . , 81Q20 ai2QQ2 294912 



for the third root, x = _„^688 4-212992 = +98304 = " 3 

 14. Example 4. Let the equation be 



a;' — 7^;* + i8a; — 18 = 0. 

 Here A = — 7; B = + 18; C=:— 18. Therefore, 



M = + 7 



?» =r — 72 

 « = — 26 



0.= + 72. 



So that we have here Cafe II. 



— 72 — ^6 



^ — V216 — 3/6 



— 26 



— —'3 



v^2i6 3/6 



r = + -^ 



3/1— T _ 3/^4v'6-— 73 _ 2\/(J— t 



Therefore, a/j^ _ Vj4/6+;3 - zv/^ + x 



3/1 T 



~vr+; 



Hence z = 



+ N/f 



■T 2 v^6 



I+T 

 .^6 , I 



and a; - 3^/6 -^^y/S _ -7^+ 3 _ zl^? _ 4 , 

 -__i3_,_l_ - —26+3 — -23 - ^ 3- 



And 3 is the only root of the equation. 



15. When t is a furd as -A7:, the value of z (in Cafe II.) al- 

 ways involves radicals of this form, Xj ''- J'~7 'i out of which 



the root may fometimes be extra<5led ; and fo the value of z will 

 be expreifed by a furd of the fame kind as r. 



Part I. P The 



