How Do You Solve for Eigenvalues of This Specific 3x3 Matrix?

[Jalo]

Homework Statement

Given the matrix A=

1 -1 -1
-1 1 -1
-1 -1 1

Find the eigenvalues.

Homework Equations

I = identity matrix

The Attempt at a Solution

det(A-xI) = (1-x)³ - 2 - 3*(1-x) = 0 ⇔
⇔(1-x)³ - 3*(1-x) - 2 = 0

I can't find a way to solve this equation...
Any help would be appreciated!4

Daniel

 

[Mentallic]

If you let y=1-x, then the cubic $y^3-3y-2$ has a zero at y=2.