Question

If equations ax³ + 2bx² + 3cx + 4d = 0 and ax² + bx + c = 0 have a non-zero common root, then prove that (c² − 2bd) (b² − 2ac) ≥ 0. 

Answer 1

Let α be the non-zero common root. So,

aα³ + 2bα² + 3cα + 4d = 0 ....(1)

aα² + bα + c = 0 .....(2)

Equation (1) − αEq. (2) gives

bα² + 2cα + 4d = 0 ....(3)

By Eqs. (2) and (3), we have