Why add the two equations ?
Why not simply subtract one from the other ?
Suppose that the first equation has roots p and q and that the second equation has roots p and s,
so that the common root is p.
Then we have
\(\displaystyle (x-p)(x-q)=0 \text{ and }(x-p)(x-s)=0\; .\)
Subtracting the second equation from the first,
\(\displaystyle (x-p)\{(x-q)-(x-s)\}=0\; ,\)
so
\(\displaystyle (x-p)(s-q)=0\; ,\)
showing that x = p is the root of the resulting equation, \(\displaystyle (\;s \neq q\;)\;.\)
So, going back to the original equations and subtracting one from the other,
\(\displaystyle x(a-c)+(b-d)=0\;,\)
\(\displaystyle x=\frac{d-b}{a-c}\; .\)
Tiggsy