Please find 'm' so that this equation has four roots which form an Arithmetic Progression???

Question

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x^4-(3m+2)x^2+m^2=0

Answer 1

Because you only have even powers of x, the roots will be
-p, -q, +q, +p
Since p-q = q - -q
p = 3q
So roots are -3q, -q, q, 3q
Thus the equation is (x^2 - q^2)*(x^2 - 9q^2)
= x^4 - 10q^2 + 9q^4 = 0

Clearly 3q^2 = m
10q^2 = 3m + 2
So 10/3 * m = 3m + 2
m = 6

The roots are -sqrt(18), -sqrt(2), +sqrt(2), +sqrt(18)
with a difference of 2*sqrt(2) between them