<2UADRATIC E<5UATI0NS. 357 



3. Add the square of half the coefficient of the second 

 term to both fides of the equation, and that side, which 

 involves the unknown quantity, will then be a complete 

 square. 



4. Extract 



^-f — is greater than fL, the square root of^4-fL (-y/3-|-f_) 

 4 4 4 4 



will l?e greater than V' — » or its equal — j and consequently 



^— ./ ^-f — -f — is always a negative quantity. Theiefore, 

 4 2 



yhen x*-T-<ix=:^, we shall have :>f=-|-|/ ^-f- f_ -j forthi? 



' jj^rmatlve value of y, and xrr — y' ^4 4 for the nef^a^ 



^ive value of .v^ 



.a' , . ^7 



la the third form, where 5:= -y/-- — h -| , both the values 



4 ^ 



pf ^' will be positive, supposing — is greater than h. For the 



4 



first value, viz. xz=.'\'a/- 3-{- — , is evidently afHrmative, 



4 ^ 



being composed of two affirmative terms. The second value, 



viz. x= — -/ h A , is also affirmative ; for since — is 



4 2 4 



greater than 1^ therefore */ — or — Is greater than 



*/" ii and consequently — y' — — ^ 4 'wIII always be 



an affirmative quantity. Therefore, when «*— ^xn: — h^ we shall 



have 



