Second Order Linear Difference Equations with constant Coefficient - The Homogeneous

Hi,

I have a problem with part of this example - "Solve the difference equation , subject to initial conditions "

I know what the characteristic polynomial is, how to calculate its zeros, but when it comes to solving and I find it difficult. I understand that we can present and than we can calculate , but how do we calculate and ? How can I establish that ? I would be greateful if anyone could help me. Thanks in advance.