82 Algebra. 



/. Giveii Vq + ^"+-^==11. (i) 



Transpose everything but the radical to the right-hand side of the 

 equation and we obtain 



\/g-{-x=Ti~-x. (2) 



Squaring both sides gives 



9-|-.t=I2I — 22X-f-x' (t^) 



and solving this quadratic we find 



x=7 or 16. 

 From (2) to (t^) is a questionable derivation ; for squaring both 

 members of an equation, 7. = R, we have found (^Art. 10) to be 

 eqtiivalent to multiplying through by L-\-R, and that the result- 

 ing equation is equivalent to the two equations 



\L=R \ 



Therefore (2^) is equivalent to the two equation 



or to 



j — V9+-^-|--^=ii j ^^ 



Hence, if we understand equation (^ij to read 



The positive square root of ((^-\-x)-\-X'=\\ 

 then a new solution has been introduced between (2) and (-t^). 

 But if we understand equation f i j to read 



A square root of (g-j-x)-\-x= 1 1 

 then it is equivalent to both the equations in (^), and no solution 

 has been introduced. This is because the introduced equation, 

 ±/.-f /v'=o is identical with the original equation -^L—R. 



In these cases the student will always find that rationalization 

 may or may not be considered as a questionable derivation acco}ding 

 as we consider the radicals to call for a particular root or A'NY root 

 of the expressions involved. 



It is more in accordance with the generalizing spirit of algebra 

 to consider the radical sign, wherever it occurs, as calling for any 

 of the possible roots. This will be better appreciated by the stu- 

 dent when he learns in Part II that every expression has three 

 different cube roots, four fourth roots, five fifth roots, etc. 



