Quadratic Eqn theory WBJEE1975

Let say that α and β are the roots of the equation ax²+bx+c=0.

Then, α + β = -b/a .....(1) and αβ = c/a.......(2)

From (1) & (2)

c/b = -αβ / α + β .............(3)

Now we have the relation between the coefficient of the polynomial as:

b³+a²c+ac²=3abc

⇒ b³ = 3abc - a²c - ac²

⇒ 1 = (3abc - a²c - ac²)/ b³

⇒ 1 = ac (3b -a - c) / b³

⇒ 1= (a/b)(c/b) (3-(a/b) - (c/b))

Sunstituting the values from equation (1), (2) and (3). The equation reduces to

(α+β)³ = αβ [3 (α+β) + αβ + 1]