Question

Suppose that the Quadratic equations ax2 + bx + c = 0 and bx2 + cx + a = 0 have a common root. Then show that a3 + b3 + c3 = 3abc.

Answer 1

Let α be the common root of the equations ax2 + bx + c = 0 and bx2 + cx + a = 0

∴ aα2 + bα + c = 0

bα2 + cα + a = 0

(bc – a2)2 = (ab – c2) (ac – b2)

b2c2 + a4 – 2a2bc = a2bc – ab3 – ac3 + b2c2

a4 + ab3 + ac3 = 3a2bc

a(a3 + b3 + c3) = 3a2 bc

a3 + b3 + c3 = 3abc (∵ a ≠ 0)