Find the condition on `a , b ,c ,d` such that equations `2a x^3+bx^2+c x+d=0` and `2ax^2+3b x+4c=0` have a common root.

Explanation

Let `alpha` be a common root of the given two equations. Thus
`2aalpha^(3) + balpha^(2) + calpha + d = 0` (1)
`2aalpha^(2) + 3balpha + 4c = 0` (2)
Multiplying (2) with `alpha` and then subtracting (1) from it, we get
`2balpha^(2) + 3calpha -d = 0` (3)
Now, Eqs. (2) and (3) re quadratic having a common root `alpha `, so
`(-2ad - 8bc)^(2) = (-3bd - 12c^(2)) (6ac - 6b^(2))`
`rArr (ad + 4 bc)^(2) = (9)/(2) (bd + 4c^(2) (b^(2) - ac)`