Let $\mathbf{A} = \begin{pmatrix} 1 & 5 \\ 1 & 3 \end{pmatrix}.$  Find all real numbers $x$ such that that matrix $\mathbf{A} - x \mathbf{I}$ is not invertible.  Enter all the solutions, separated by commas.
<|im_start|>system
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<|im_start|>user
Let $\mathbf{A} = \begin{pmatrix} 1 & 5 \\ 1 & 3 \end{pmatrix}.$  Find all real numbers $x$ such that that matrix $\mathbf{A} - x \mathbf{I}$ is not invertible.  Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2 + \sqrt{6}, 2 - \sqrt{6}
Let $\mathbf{A} = \begin{pmatrix} 1 & 5 \\ 1 & 3 \end{pmatrix}.$  Find all real numbers $x$ such that that matrix $\mathbf{A} - x \mathbf{I}$ is not invertible.  Enter all the solutions, separated by commas.
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