220 MB. K. HABLEY ON IMPOSSIBLE 



The equation (a) is possible or impossible according as —3 

 (jac -|- bd) is positive or negative, and (/3) is always possible. 



When —3 (ac -j- bd) is positive, (a) has one root, viz., 



a: z= ^ (6 4- 4ai — >/& -\- 4.abd + 4a'c), («')' 



and (;S) has one root, viz., 



x=~.{b -f 4a<f + V&' 4- 4:abd + 4^ (^')5 



when -5(ac -f- ^^) is negative, (a') and (j^) are both roots 



of (/3), and (a) is impossible. 



These are useful criteria, as they enable us by mere in- 

 spection to ascertain a priori, whether any proposed irra- 

 tional equation of the form (a) or (/3), having numerical 

 co-efficients, be possible or impossible ; and likewise to de- 

 termine, when the equation is possible, which of the roots, 

 furnished by the usual process of solution, do, and which 

 do not belong to the proposed equation. 



Thus, let the equation 



Sx -f- V30ar — 71 — 5 

 be proposed. (See Art. o.) Then, since this agrees in 

 form with (a), and in this case — 5 (ac-^-bcl) is negative, 



we at once pronounce it to be impossible. The roots 4 



8 

 and q , given by the usual method of resolution, belonging 

 o 



to the corresponding equation 



3x — V30x — 71 =r 5. 

 In practice, perhaps, it is more convenient to place the 

 equation under the form (ai) or (/3,). Thus, the equation 

 above proposed may be written as below : — 



(3j: — 5) 4- VI (3a; —TsT^^l =: 0, 

 and since the last term under the radical is negative, we 

 conclude immediately that the equation is impossible. 



