Q.385. Polynomial equation with coefficients in A.P.

Question 385.
Find a necessary and sufficient condition on a, b, c
such that the roots of x³ + ax² + bx + c = 0 are in arithmetic progression.

Answer 385.
Let the roots be m-d, m and m+d
=> m-d + m + m+d = - a
=> m = - a/3
m(m-d) + m(m+d) + (m-d)(m+d) = b
=> 3m^2 - d^2 = b
=> d^2 = a^2/3 - b
=> roots are
-a/3 - √(a^2/3-b), - a/3 and -a/3 + √(a^2/3-b)
=> product of the roots
(-a/3 - √(a^2/3 - b) * (-a/3) * (-a/3 + √(a^2/3 - b) = - c
=> - (a/3) * (a^2/9 - a^2/3 + b) = - c
=> (a/3) (b - 2a^2/9) = c
=> c = (a/27) (9b - 2a^2)
This is the necessary and sufficient condition for the roots to be in A.P.
Sufficient because the roots of the equation with the above value of c are the ones found as above for which refer to the following Wolfram Alpha link:
http://www.wolframalpha.com/input/?i=x%C2%B3+%2B+ax%C2%B2+%2B+bx+%2B+%28a%2F27%29%289b-2a%5E2%29+%3D+0

Link to YA!