////: >//; i/'./// i.i.yg 111 



two straight lines are here represented by ah equa 

 f the tecond degree. 



Conversely, if an eqn the second degree, whose 



eron.l : in ii.r. member separated 



nt degree factors, with real coefficients, an in 



-.jil.i! : :. ( ' .. thru its I.M Mis * ..liMsls of two HtHUght 11 DCS 



Thus tl- equation 



VJT + ! + 7 - 



may bti written in tin- fnnn 



-3y + 7x* + y + U=-0, 



which nhows tli lied when '2x - *y 4- 7 = 0, and 



also when .r + y + 1 = 0. Its locus is tln-refore composed 

 of the two lines whose equations are : 



Jr-8y -f 7 = 0, umi-r + y -|-1=0. 



67. Condition that the general quadratic expression may be 

 factored. 1 he most general <M|uuti<>n ,,f the sffond degree 

 U-t \\ITII tun variahlea may be written in the form 



// , + Bf + Gx + Fy+ e=0. . (1) 



required to tin-l tli> tiiat must exist among the 



OotifficienU of this equation in .nh -r th.it its first member 

 may be w? part UM! in' ;)M- first 



degree, i i-.-quiiv.! t tin<l tin- (ondition that the equa- 



may be written thus: 



*># * 'i)0v * V + r i) = - 



;tly if equnt . can be written in the form of 



equation (_'). tli.-n tin* valnrs of x obtained from equation 

 are rational, and are either 



or ,--!-**. 





