66 AlvGKBRA. 



3. By solving the equation 

 it will be found that its roots are 



-ie+ sfV-f and -\e-sj\e-f. 



4. ThEORKm. Every quadratic fimdion of x can be resolved 

 ijito the product of two linear functions of x. 



Take the function of x in the form 



x'-^ex-^f. 

 add and subtract \e:^ from the function, thus not altering its value. 

 We obtain then 



x^J^ex-V\e'-\e-\-f 

 This may be v^^ritten 



(x^\er-(Y-f), 

 or, if we please, as the difference of two squares, 



{x-^\ey-{s/\^^)\ 

 Writing this as the product of the sum and difference, it takes 

 the form 



[(A'4-i^)- Vi?^] [c^+i^)+ vv=7] 



or (x^\e-sf\e^-f)(x^\e-^ sf V-f). 



which is the product of two linear functions of x. 



5. ExAMPLKS. Resolve the following quadratic functions into 

 the product of two linear functions of x : 



1. x'—x—2\o. 



2. 3jr^H-2a'— 85. 

 J. x'^—bbx-\-(^b-. 

 ^. /^a" X- — \ax -\- \ . 

 5. .r^-i4.r4-33. 



6. Theorem. If the roots of a quadratic equation are a and b, 

 then the equation may alivays be put i?i the form (x—a)(x—b)=^o. 



By Art. 4, the equation 



x''+ex-\-f=o (i) 



may always be placed in the form 



{x+\e^-sl\e^-f){x-^\e- s/\^f)=o. (2) 



