Find Eigenvectors of 3x3 Matrix

[0specificity]

Homework Statement

Find the eigenvalues and an orthonormal set of eigenvectors for this matrix:

|1 1+i 0|
|1-i 1 0|
|0 0 2|

Homework Equations

Find the determinant of A - xI, where A is the matrix, I is the identity matrix, and x denotes eigenvalues
Set the determinant equal to 0, and then find eigenvectors for each eigenvalue

The Attempt at a Solution

If x denotes an eigenvalue, I found the determinant of this matrix to be (2-x)((1-x)(1-x) - (1+i)(1-i)) = (2-x)(x^2 - 2x - 1). Then I got stuck, since I do not know how to factor this equation. How do I find the eigenvalues for this matrix? Am I doing something wrong?

 

[Dick]

You've already factored it as much as you need to. What's left is a quadratic equation. You can solve that without factoring, can't you?