If a,  b, c  are in GP., the equations  ax2+2bx+c=0  and dx2+2ex+f=0  have a common root if d/a,  e/b, f/c  are in:

Since a,  b, c  are in G.P., ∴       b2=ac    so  ax2+2bx+c=0 has equal rootssince Δ=4b2−4ac=0 and the root is −2b/2a=−b/a Therefore the common root must be −b/a .∴      db2a2−2eba+f=0 or   daca2−2eba+f=0 or    da−2ebac+fc=0              or          da+fc=2eb i.e., d/a,   e/b,  f/c  are in A.P.