188 BOOK OF MATHEMATICAL PROBLEMS. 



889. The roots of the equation x*-px* + qx-r = are in 

 harmonical progression ; prove that those of the equation 



n (pq - n'r) x* - (p* - 2npq + 3n*r) x 3 + (p q - 3nr) x-r = Q 

 are also in harmonical progression. 



890. Reduce the equation x a px* + qx r = to the form 

 2/* =b 3y -f m = by assuming x = ay + bj and solve this equation 



by assuming y = z =F - . Hence prove that the condition for 



equal roots is 



4 (p* - 3q) 3 = (2p* - 9pq + 27r) 9 . 



891. In solving a cubic by Cardan's rule; if a, /?, y be the 

 roots of the complete cubic, the roots of the auxiliary quad- 

 ratic are 



64 

 892. Solve the equations 



- 2) 8 + 3 = 4z (3x* + 4), 



893. If a, ft y, 8 be the roots of the equation 



x 4 + qx* + rx + s = 0, 

 the roots of the equatipn 



V + qa(l -8)*x> + r (1 -s) 3 x + (1 -s) 4 = 



Ul be ^ + y + 8 + 



894. In the equation 



a; 4 -px 3 + qx 9 - rx + 8 = ; 



prove that the sum of two of the roots will be equal to the sum 

 of the other two, if Sr 4pq + p 3 = ; and that the .product of 

 two will be equal to the product of the other two, if p 9 s~ r*. 



