1818.] A new Quadratic Theorem. 35 



For the mediate terms a x doubled = 2 a x and + 26 

 squared = 4 6 2 , which subtracted leaves nothing. 



Extract the square root of 



a 2 - 8 a* b I + 2 a b* + 16 a* b* I - 8 a b 3 +. b* 

 a -4a 4| |+ 6»+0 



For the mediate terms + a x + b a - = a 6 2 , which doubled 

 = 2 a 6 2 , and — 4 ab squared = 16 a 2 6% which subtracted 

 leave nothing. 



Extract the square root of 



a 4 - 2 a 3 

 a 2 — a 





+ 3 



— 2 a x 3 



+ x* 



+ x* 

 + 



For the mediate terms u* x x* = + a 2 # 2 , which doubled 

 = 2 a* a 2 , and — a # squared = + a 2 x 2 , which added 

 = + 3 a 2 j 2 . 



Extract the square root from the following hexanomial : 



a 2 

 6 



6 I — 16 a 2 + 4 6 s 



+ 



32 6 



8 



+ 

 + 



64 

 



a 4 — 4 

 a 2 - 2 



For the mediate terms + a 2 x — 8 = — 8 a 2 , which doubled 

 = — 16 a 2 , and — lb squared = + 4 6 s . 



Some mathematicians, sensible of the tediousness by evolu- 

 tion, have advised the extraction of the roots of the most simple 

 terms, connecting them together by the signs + or — , as may 

 be judged most fit, then involving this compound root to the 

 proper power ; if it be the same with the given quantity, the 

 square root is found as in the following example : a 2 + 2 a b + 

 2ax + b*+2bc+c' i . According to this irregular method, 

 like sailing without a compass, the first, fourth, and sixth terms, 

 are supposed to give the square root ; whereas in reality it is 

 extracted from the first, second, and fifth terrns, as may be 

 seen according to the following working : 



a + b\ j + c + 



For the mediate terms a x c = a c, which doubled = + 2 a c, 

 and b squared = + 6 2 . 



Some may prefer the following method of working these sums. 

 Extract the square root from 



+ « 4 

 - 2 a 3 b 



+ 2 a'' 



-2ab 

 + b< 



+ 3rt 2 6 2 



- 2ab 3 

 + b< 



« 2 

 ^■ab 



+ b* 







c 2 



