Question

the condition that the roots of ax ³+bx²+cx+d=0 may be in A.P is 2b³+27a²d=9abc

Answer 1

Answer:

Let the roots be p,q,r. The roots are in A.P.

So,

2q=p+r,

3q=p+q+r.

Using theory of equations,

p+q+r= -b/a .

So, q= −b/3a .

But, q is a root of the given equation. So

aq 3+bq 2 +cq+d=0,

a( -b/3a ) 3 +b( −b/3a ) 2+c( −b/3a)+d=0,

Rearranging, we get

2b*3 +27a*2d=9abc.