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]