1G BOOK OP MATHEMATICAL PROBLEMS. 



84. Obtain the several equations for determining a, /?, y 

 that the equations 



x 4 + 2px* + qx* + rx + s = 0, (te 8 -f px + a) 8 = (/3x + y) 8 

 may coincide : and in this manner solve the equation 



85. The roots of the equation 



(x + a c) (x + b + c) (x + a d) (x + b + d) = e 

 will all be real, if 



86. If x l , x 2 be the roots of the equation 



(l-7/i)(l -x) mx 



then will - - r- - + b + - = 0. 



87. If y, z be the roots of the equation 





bc ca ab 



tUeu.il! - 



b c ca ab 



88. Prove that the equation 



x 3 + ax 3 + bx + = 

 3 a 



can be solved directly, and that the complete cubic 



may be reduced to this form by the substitution x = y + h. 

 Prove that the roots of the auxiliary quadratic are 



a, ft y being the roots of the cubic. 



