Your equation is:

Assume this has four distinct real roots (as it must if they are to be in arithmetic progression since it can only have 0, 2 or 4 real roots and so you need 4 to have an arithmetic progression), and as this is a quadratic in , they are of the form: .

So for these to be in arithmetic progression means that:

where and are the distinct positive roots of and is the smaller of the two.