I believe there's a little trick involved here:

Let

(1)

(2)

Observe that is not a root of (2) for any choice of b. Hence this can not be a common root.

So we may replace the problem of finding b such that (3),(4) have a common root

(3)

(4)

Then a common root of (3),(4) must also be a root of the difference of (3),(4)

This implies that is the only possible common root they can have.

substituting in (2) gives 2 possible values for b.