Diagonalizing a (dimensionless) Hamiltonian

[Cocoleia]

Homework Statement

Diagonalize using creation / annihilation operator methods

Relevant Equations

a = 1/sqrt2 (y+dy)
a- = 1/sqrt2(y-dy)

I am given this Hamiltonian:

And asked to diagonalize.
I understand how we do such a Hamiltonian:

But I don't understand how to deal with the extra term in my given Hamiltonian. Usually we use

To get

 

[DrClaude]

Isn't it simply that you need to express $x$ in terms of $a$ and $a^\dagger$?