Show that \(( \alpha - \beta ) ^ { 2 } \equiv ( \alpha + \beta ) ^ { 2 } - 4 \alpha \beta\).
The quadratic equation \(x ^ { 2 } - 6 k x + k ^ { 2 } = 0\), where \(k\) is a positive constant, has roots \(\alpha\) and \(\beta\), with \(\alpha > \beta\).
Show that \(\alpha - \beta = 4 \sqrt { 2 } k\).
Hence find a quadratic equation with roots \(\alpha + 1\) and \(\beta - 1\).