2011/103) Find sufficient condition on a, b, c that the roots of x³ + ax² + bx + c = 0 are in arithmetic progression.

Let the roots be m-d, m and m+d

then as per vieta's formula http://en.wikipedia.org/wiki/Vi%C3%A8te%…
1)
sum of roots = m-d + m + m+d = - a or 3m = - a
2)
b= m(m-d) + m(m+d) + (m-d)(m+d) = 3m^3 - d^2 or d^2 = 3m^2 - b or (a^2)/3- b
3)
product of roots = m(m-d)(m+d) = m(m^2 - d^2) = - c

or (-a/3)((a/3)^2 - (a^2/3 -b) = -c

or(- a/3)(2a^2/9 - b) = - c

or (a/3) (b - 2a^2/9) = c

or (a/27) (9b - 2a^2) = c
above condition is necessary and sufficient