20 4 On the RESOLUTION of 



Case III. When BB — 4 AC is a fquare. 



b c 



Let x 2 -\-—x -f- — zz DXE \ then the divifors of ax"--\-bx-\-c 



will be — XD, and »XE. But it appears, from the doctrine 

 of equations, that the excefTes of x above the roots of the 



be 



quadratic, x 2 -\-—x -f- — 2= o, are the divifors of the expreflion 

 x % 1 — x _| Wherefore, D = #-| , and E zz 



1 a 'a ' ' 2a ' 



^_j_ _ # Hence, when V{b 2 — ^ac) is a whole or fractional 



number, the expreflion ax 2 -\-bx-\-c admits of refolution, and 



the divifors are— \x-\ — J and n{x-{- — ) • And 



when thefe are found, the folution will be obtained from 

 Prob. VI. 



Suppose 14^+ 19*+ 6 == y 2 , then b 2 — ^ac zz 361 — 336 =1 



25, and d = ^ (*+ ^r - )' and E = ; <^+^r-)- If » = 2 > 



the divifors will be — (x+— ) == jx-\-6 t and 2(#+— ) = 



■zx-\-i 5 whence, from Prob. VI. # zr 2, and y zz 10. For 

 14.4+19.2+6 == 100 = (ioo)\ 



Case IV. When the general quadratic can be refolved into factors, 

 if diminijhed by a given fquare. 



Let (ex-\-f) (gx+b) zz y 2 — d 2 t then (ex-\-f) (gx+h) zz 



y+d 



{y-\-d){y — d) j whence ex-\-f zz- my — md 9 and gx-j-h zz ' m . 



By 



