204 0« the RESOLUTION of 



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



h c 



Let x^-\ X -\ = DXE ; then the divifors of ax^-\-bx-\-c 



will be — XD, and ;/XE. But it appears, from the dodrine 

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

 quadratic, x''-\ x + — = o, are the divifors of the expreflion 



,v'+ — X + — . Wherefore, D =z x-j , and E — 



x-\- — "^'' . Hence, when V'{b^ — ^ac) is a whole or fradional 



number, the expreflion ax'-j-dx-j-c admits of refolution, and 



the divifors are— (jcH J and kU+ ) And 



when thefe are found, the folution will be obtained from 

 Prob. VI. 



Suppose i4.r"+i9v+6 =: y\ then b^—^^c — 361—336 =: 



25, and D = -^ ix+ '-^f), and E = nix+'-^^). If ;; iz 2, 



the divifors will be -7-Gr+—) =: yx+e, and 2{x-i-—) — 



2x-\-i ; whence, from Prob. VI. x — 2, and y — 10. For 

 14.4+19.2+6 = 100 — (100)". 



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

 if diminifhed by a given fquare. 



Let (ex-^-f) {gx-\-h) =z y'—d\ then (.ex+f) igx-^h) = 



By 



y+d 



{y-\-d){y—d) ; whence ex-\-f =■ my — md^ and gx-\-h — ^-^ 



