Young — Product of the Squares of the Differences, etc. 747 



the squares of wMcli differences are 



1-, 3-, 2^ ; and the product of these is 36 ; 

 and the former result, when its sign is changed, is the same number. 



2. The roots oi s(? - 4x' + 3x + 2 = are 



2, 1 + A 1-V2; 



and the expression Q" - 3PP' is 



^" + Bx+ C ^ 1x' - ZQx + 33 ; 



.-. ^2 = 900, and 4^ C= 924; 



.-. B""- AA 6' = - 24 ; and ^^ = - 8. 



3 



The differences of the roots are 



1-V2, 1+V2, 2V2; 



and the product of these differences is - 2V2, of which product the 

 square is 8 ; the same as the former result when its sign is changed. 



3. It may be well to work out this final example in more detail : 



i' = ^+ lire'- -102:» + 181 = 

 Q = Zx' + 12x-\02 

 P'= 3:c+ll. 



Multiplying Q by itself, and omitting terms in x^ and ^*, 

 3a;2 + 22^-102 

 3^2 + 22^-102 



AMx^ - 612a;2 - 448 8x + 10404 

 .-. Q2 = - 128^2- 4488a; + 10404. 

 In like manner, multiplying P by P', 



^+11a"-102^+ 181 

 3^+11 



121^- - 306,r - 1122.2; + 543a; + 1991 

 = _ I85x^ - 579a; + 1991 = PP' 

 .-. Q--3PP'= 427a;- -2751a; + 4431 =Ax' + Bx+ C ; 

 .-. B' = 7568001 

 4^(7=7568148 



447 = ^ -4^ C. 



