AP EAPCET 19-Aug-2021 Shift 2 Paper

Let α be the common root of
x3+ax+1=0
and x4+ax2+1=0
Then,α3+aα+1=0
and α4+aα2+1=0
subtracting both equations,
α4−α3+aα2−aα=0
α(α3−α2+aα−a)=0
⇒α2(α−1)+a(α−1)=0 {α=0 doesn'[ give a root]
⇒(α−1)(α2+a)=0
⇒α=1 or a=−α2
Clearly, α=1 gives a=−2 in both equations.
Hence, for a=−2 both equation have a common root.