58 Algebra. 



26 . x" — I = k(kx^ — 4.x— k) . 



27. (x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a)=o. 

 Result, x=^\(a-\-b-^c)-^\s/ar-\-b'-^c'—ab—bc—ca. 



28. X' H- 6a' 4- 2 1 = I o. 



This becomes, in the typical form, 



jr^ + 6;f= — II. 

 Completing the square, we obtain 



jf^-f 6-r+9= — 2. 

 Now we cannot obtain the square root of the right-hand mem- 

 ber of this equation ; for it is a negative quantity, and the square 

 of no algebraic number can be negative. But, if we were to go 

 through the operation of finding x as has been done in the other 

 cases above, and indicate the root of —2 as if we could take it, 

 we would have 



jr= — 3zbv/— 2. 

 Thtis we have had forced upon us in the solution of the quad- 

 ratic equation, something which, whatever interpretation it may 

 have, is evidently 7iot a?t algebraic quantity in the sense in which 

 the term is commonly used. Such an expresion is called an im- 

 agijiary, and its treatment is reserved for Part II of the pres- 

 ent work. In the next chapter will be found a discussion of the 

 circumstances under which such expressions occur. 

 2g. 4-r^-|-4-r+4=jf^. 



13. Problems Requiring the Solution of Quadratic 

 Equations. The student in his previous study has probably 

 already noticed that the first task in the algebraic solution of a 

 problem is always an attempt to express the language of the prob- 

 lem in algebraic symbols ; that is, to cast the relations and condi- 

 tions expressed by the words of the problem into an equivalent 

 statement in the form of one or more algebraic equations. This 

 work is called the statement of the problem, and is generally a 

 difficult one for the beginner to perform. When the statement of 

 a problem is complete, all that remains to be done is the solution 

 of the equation or equations obtained thereby by processes already 

 familiar. 



We wish to strongly emphasize the fact that the equation 

 obtained by the translation of the words of most of the algebraic 

 problems in the books is often not an exact equivalent to the condi- 



