The quadratic equation \({x^2} - 2kx + (k - 1) = 0\) has roots \(\alpha \) and \(\beta \) such that \({\alpha ^2} + {\beta ^2} = 4\). Without solving the equation, find the possible values of the real number \(k\).

\(\alpha  + \beta  = 2k\)    A1

\(\alpha \beta  = k - 1\)    A1

\({(\alpha  + \beta )^2} = 4{k^2} \Rightarrow {\alpha ^2} + {\beta ^2} + 2\underbrace {\alpha \beta }_{k - 1} = 4{k^2}\)    (M1)

\({\alpha ^2} + {\beta ^2} = 4{k^2} - 2k + 2\)

\({\alpha ^2} + {\beta ^2} = 4 \Rightarrow 4{k^2} - 2k - 2 = 0\)    A1

attempt to solve quadratic     (M1)

\(k = 1,{\text{ }} - \frac{1}{2}\)    A1

[6 marks]