Question 103
Find all values of the real parameter $a$ such that the equation
$$
x^3 - a x^2 + (a^2 - 1)x - a^2 + a = 0
$$
has three distinct real roots which (in some order) form an arithmetic progression.
$$
x^3 - a x^2 + (a^2 - 1)x - a^2 + a = 0
$$
has three distinct real roots which (in some order) form an arithmetic progression.