AJfl) OTIIEK SURD EQUATIONS. 211 



multiplying the equation by 4 -f- Va^ — 8 — V^; -j" 21> for 

 it satisfies the equation 



4 -4- -v/a* — 3 — VicH-2T zz 0. 

 For the sake of further illustration, let the particular 

 example 



3a: + V30a; — 71 =: 5 

 be proposed. Eliminating the radical sign by the usual 

 method, we get 



9x2 __ gox + 96 — 0, 

 which resolved, gives i» rr 4 or f . Now, neither of these 

 values, when substituted for x, are found to satisfy the pro- 

 posed equation ; the equation really satisfied by them is, 



3a; — VSOir — 71 =r 5. 

 We therefore conclude that 



3a; -f v/30a; — 71 = 5 

 is an impossible equation. 



In Wood's Algebra by Lund (thirteenth edition), page 

 128, the equation we are now considering is discussed. 

 After noticing the inadmissibility of the two values of x, 

 above found as roots of the equation, the able Editor 

 remarks, '' whether there be any values of ^ or not, which 

 will satisfy the equation 3 .r + V^Oo; — 7 1 n: 5, we can- 

 not say ; all that we know is, that the common method of 

 solution will n,ot produce them." From what has been 

 above shown, it is manifest that the doubt which is here 

 expressed as to the possUnlity of the proposed equation, is 

 altogether without foundation. It assumes, in fact, that 

 the same value which satisfies the irrational equation does 

 not necessarily satisfy also the rational one; but this 

 assumption we know to be false. (Art. 3.) 



6. In the algebraic solution of a certain class of problems, 

 it is often of considerable importance to know a priori, 

 whether the irrational equations which express the given 

 conditions be possible or impossible, and (if possible) to 



