ax²+bx+c=0

The solutions are

 and  


 cy²+by+a=0

The solutions are

 and 

They are reciprocals then their product must be 1:

Let's multiply the solution of the first with a positive radical times
the solution of the second with a negative radical:

 





The two factors in the numerators are conjugates so FOILing them
causes outers and inners to cancel, so we get:

 =

 =

 =

 =

1

so they are reciprocals.  Multiplying the other pair
gives the same results.

Edwin