70 Algebra. 



This is an equation of the first degree in terms of y\ Solving 

 we obtain 



whence j/= ± V \e' — /, 



and since x=v—y^, 



Solve in this manner the equation j^:'— 5Ji:--i4=o. 



(b). By considering the quadratic as the product of tivo linear 

 factors. 



Suppose the function of x to be the product of two factors of 

 the form (x-\-\e-{-2i)(x-\-\e—ii), where n is a new unknown 

 quantity. Then we have the equation 



x^^ex+f=(x+\e+u)(x+}^e-u). 

 Expanding the right member of the equation we obtain 



X- -\- ex +/= x^-\-ex-\- \e^ —ic". 

 Therefore u'=^\e-—f, 



whence u—-±zs/ \e' —f 



Now as the product of the two factors (x-\-\e-\-7i)(x-^\e—u) 

 must equal zero we must take x either —^e—u or —\e-\-ii. Take 

 the former, and 



-^ — —\e—ii=— U^ V i^' — /. 



Solve by this method the equation .r'^— 6^=1 6. 



(c). By the sum and product of the roots. 



Suppose the two roots of .r^ + <?'-r+/=o to h^ y and z. Then we 



know by Art. lo, 



y^z=^—e (l) 



and J'-^=f, (^) 



squaring (i) we obtain 



y''-]r2yz-\-z^—e\ (2^) 



Subtracting four times (2) from this 



y^2yz^z^=e^-jj, 



or, extracting the root, r— 2'=±\/^''— 4/, • 

 and since y-\-z=—e, 



—e^sje—dj 



2 

 Solve in this manner the equation 3:1-- — 5-r+2=o. 



