Quadratic Equations. 



59 



tio7is and relatio7is told in the lufiguagc of the problem. In fact, 

 the equation often efnbraces more than the problem itself. We 

 will illustrate this by the following problem : 



A certain number consists of two digits whose sum is lo. If 

 we reverse the digits and multiply this new number by the 

 original number, the product will be 2944. Required the number. 

 I^et x=the digit in unit's place ; 

 then 10— -r=the digit in ten's place, 

 and io(^io— .rj=the value of the digit in ten's 

 place ; 



whence \o( \o— x ) -\- x=\h& value of the orig- 

 inal number ; 



also lox-f (^10— .rj=the value of the number 

 with the digits reversed. 

 But, by the problem, 

 [io('io— .i-;-f .r] [ioi--f ("lo— .i-;]=2944. (\) 



Statement, or trans- 

 lation of the language 

 into an algebraic equa- 

 tion. 



! That is, 



Solntiuii of the cciiui 

 tion. 



2944. 



(2) 



(^ I GO — 9 Ji" j (^ I o -f g.r j : 

 Expanding left member, 



8 IX''— 8 lox— 1000= — 2944. 

 Transposing and uniting, 



8 1 Ar — 8 1 ox= — 1 944. 

 Dividing through by 81, 



x^— ioa:= — 24. 

 Completing square and solving, 

 jt==4 or 6. 



The number is therefore either 46 or 64. Now consider equa- 

 tion ('i j as a translation of the problem into algebra. As far as 

 is stated by the equation ( i ) the unknown quantity x may be any 

 algebraic quantity conceivable, — positive or negative, integral or 

 fractional, rational or irrational, or, in fact, it may possibly be 

 what we have called an imaginary. As far as the equation ex- 

 presses the nature of a', it may as likely turn out in the solution 

 one kind as another of those enumerated. But, as expressed in 

 the language of the problem, x must be a digit ; that is, a positive 

 integral number less thayi ten. The equation does not express 

 this fact and cannot be made to do it. The reason why the prob- 



