= Solution
Set $\mathbf x=\mathbf a\cos(\omega t+\varphi)$. The two <normal modes> satisfy
$$
\left[\begin{pmatrix}2&-1\\-1&1\end{pmatrix}-s\mathbf I\right]\mathbf a=0,\qquad s=\frac{m\omega^2}{k}.
$$
The <characteristic equation> is $(2-s)(1-s)-1=s^2-3s+1=0$. Hence
$$
\boxed{a_\pm=\frac{3\pm\sqrt5}{2},\qquad\omega_\pm^2=\frac km\frac{3\pm\sqrt5}{2}.}
$$
Here $a_\pm$ denotes the scalar coefficient requested in the source, whereas bold $\mathbf a$ denotes a mode-amplitude vector. Both roots are positive.
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