Pl. answer this question

α+β+γ = -3p

αβ + βγ + γα = 3q

αβγ = -r

If the roots are to be in arightmatic progression then,

α = a-d; β = a; γ = a+d

α+β+γ = 3a = -3p

a = -p

and

αβγ = (a²-d²)a =  -r

(a²-d²)a = -r         ................(1)

Put the value of a = -p

we find,

d = (p² - r/p)

Also,

αβ + βγ + γα = 3q

a² - ad + a² + ad + a² - d² = 3q

2a² + a² - d² = 3q

2a² + a² - d² = 3q

d = (3p² - 3q)

Equating the two d's

(3p² - 3q) = p² - r/p

Hence, p, q and r must satisfy this relation if the roots are to be in AP.

Regards,

Team,

TopperLearning.