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/(q−q′) = 1/(p′−p)

Then,

α²/(pq′−p′q) = α/(q−q′)     or,     α/(q−q′) = 1/(p′−p)

α = (pq′−p′q)/(q−q′)          or,          α = (q−q′)/(p′−p)

So, α be either (pq′−p′q)/(q−q′) or.  (q−q′)/(p′−p)