If the roots of the cubic equation a x^3+b x^2+c x+d=0 are in GP, then

Step-by-step solution

Let A R , A, A R be the roots of the equation a x^3+b x^2+c x+d=0 , then Product of roots aligned & A^3=- d a = & A=- ( d a )^ 1 / 3 aligned Since, A is a root of the equation. a A^3+b A^2+c A+d=0 aligned & a (- d a )+b (- d a )^ 2 / 3 +c (- d a )^ 1 / 3 +d=0 & b ( d a )^ 2 / 3 =c ( d a )^ 1 / 3 & b^3 d^2 a^2 =c^3 d a & b^3 d=c^3 a aligned