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$$\left\{ {\begin{array}{*{20}{l}}{{u_1} + {u_2} + {u_3} = 27}\\{u_1^2 + u_2^2 + u_3^2 = 275}\end{array}} \right.$$ $$\Leftrightarrow \left\{ {\begin{array}{*{20}{l}}{{u_1} + \frac{{{u_1} + {u_3}}}{2} + {u_3} = 27}\\{u_1^2 + {{\left( {\frac{{{u_1} + {u_3}}}{2}} \right)}^2} + u_3^2 = 275}\end{array}} \right.$$ $$\Leftrightarrow \left\{ {\begin{array}{*{20}{l}} {{u_1} + {u_3} = 18} \\ {u_1^2 + u_3^2 = 194} \end{array}} \right.$$ $$\Leftrightarrow \left\{ {\begin{array}{*{20}{l}} {{u_1} + {u_3} = 18} \\ {{u_1}{u_3} = 65} \end{array}} \right.$$ $\Rightarrow u_1,u_3$ là nghiệm của pt ${x^2} - 18x + 65 = 0$\[ \Rightarrow \left\{ \begin{array}{l}{u_1} = 13\\{u_3} = 5\end{array} \right. \vee \left\{ \begin{array}{l}{u_1} = 5\\{u_3} = 13\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}{u_1} = 13\\d = - 4\end{array} \right. \vee \left\{ \begin{array}{l}{u_1} = 5\\{u_3} = 4\end{array} \right.\]

$$\left\{ {\begin{array}{*{20}{l}} {{u_1} + {u_2} + {u_3} = 27} \\ {u_1^2 + u_2^2 + u_3^2 = 275} \end{array}} \right.$$ $$\Leftrightarrow \left\{ {\begin{array}{*{20}{l}} {{u_1} + \frac{{{u_1} + {u_3}}}{2} + {u_3} = 27} \\ {u_1^2 + {{\left( {\frac{{{u_1} + {u_3}}}{2}} \right)}^2} + u_3^2 = 275} \end{array}} \right.$$ $$\Leftrightarrow \left\{ {\begin{array}{*{20}{l}} {{u_1} + {u_3} = 18} \\ {u_1^2 + u_3^2 = 194} \end{array}} \right.$$ $$\Leftrightarrow \left\{ {\begin{array}{*{20}{l}} {{u_1} + {u_3} = 18} \\ {{u_1}{u_3} = 65} \end{array}} \right.$$ $\Rightarrow u_1,u_3$ là nghiệm của pt ${x^2} - 18x + 65 = 0$ $$\Rightarrow \left\{ \begin{gathered} {u_1} = 13 \hfill \\ {u_3} = 5 \hfill \\ \end{gathered} \right. \vee \left\{ \begin{gathered} {u_1} = 5 \hfill \\ {u_3} = 13 \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} {u_1} = 13 \hfill \\ d = - 4 \hfill \\ \end{gathered} \right. \vee \left\{ \begin{gathered} {u_1} = 5 \hfill \\ {u_3} = 4 \hfill \\ \end{gathered} \right.$$