If one root of ${\mathrm{x}}^2-\mathrm{x}-\mathrm{k}=0$ is square of the other, then k =

Let $\mathrm{\alpha }$ and ${\mathrm{\alpha }}^2$ be the roots of $x^2-x-k=0.$ Then,

$\mathrm{\alpha }+{\mathrm{\alpha }}^2=1\text{ and }{\mathrm{\alpha }}^3=-\mathrm{k}$

$\begin{array}{l}\mathrm{or} (-\mathrm{k}{)}^{1/3}+(-\mathrm{k}{)}^{2/3}=1\\ \mathrm{or} -{\mathrm{k}}^{1/3}+{\mathrm{k}}^{2/3}=1\end{array}$

$\begin{array}{l}\mathrm{or} {\left({\mathrm{k}}^{2/3}-{\mathrm{k}}^{1/3}\right)}^3=1\\ \mathrm{or} {\mathrm{k}}^2-\mathrm{k}-3\mathrm{k}\left({\mathrm{k}}^{2/3}-{\mathrm{k}}^{1/3}\right)=1\end{array}$

$\begin{array}{l}\mathrm{or} {\mathrm{k}}^2-\mathrm{k}-3\mathrm{k}\left(1\right)=1\\ \mathrm{or} {\mathrm{k}}^2-4\mathrm{k}-1=0\\ \mathrm{or} \mathrm{k}=2\pm \sqrt{5}\end{array}$