Question

(x - a) (r - b) + (x - b) (x - c) + (x -c) (x - a) = 0,
Show that this root is real

Answer 1

Step-by-step explanation:

Let the common root be α

⇒α4+aα2+1=0,α3+aα+1=0

Subtracting both eqns, α4−α3+aα2−aα=0

⇒α(α3−α2+aα−a)=0  (but α=0 does't give a root)

⇒α3−α2+aα−a=0

⇒α2(α−1)+a(α−1)=0

⇒(a+α2)(α−1)=0

⇒α=1 or a=−α2

Clearly α=1 gives a=−2 in both equations,

hence for a=−2 both equations have a common root