If a, b, c are positive real numbers such that the equations ax2+bx+c=0 and bx2+cx+a=0 have a common root then

Let a be the common root of the two equations Thenand aα2+bα+c=0bα2+cα+a=0Solving these two equations, we getα2=αab−c2⇒a2=ab−a2ac−b2 and α=1ac−b2⇒ab−c2ac−b2=bc−a2ac−b22⇒ab−c2ac−b2=bc−a22⇒aa3+b3+c3−3abc=0⇒a3+b3+c3−3abc=0⇒(a+b+c)a+bw+cw2⇒a+bw+cw2=0 or a+bw2+cw=0 [∵a+b+c≠0]