If the quadratic equations x^2 + px + q = 0 and x^2+ p' x + q' = 0 have a common root . Show that it must be either ( p q' - p' q / q - q' ) or ( q - q' / p' q )
Soln:
Let α be the common root of the given equations:
α2 + pα + q = 0 …(i)
and α2 + p’α + q’ = 0 …(ii)
Solving (i) and (ii) by cross multiplication,
1 p q 1
1 p’ q’ 1
Or, α2/(pq′p′q) = α/(qq′) = 1/(p′p)
Then,
α2/(pq′p′q) = α/(qq′) or, α/(qq′) = 1/(p′p)
α = (pq′p′q)/(qq′) or, α = (qq′)/(p′p)
So, α be either (pq′p′q)/(qq′) or. (qq′)/(p′p)