tính giá trị của biểu thức A = x 1 ( x 1 + 2024 ) + x 2 ( x 2 + 2025 ) − x 2 .
Giải thích
Ápdụng định líViète,tacó: \(\left\{ \begin{array}{l}{x_1} + {x_2} = \frac{5}{2}\\{x_1}{x_2} = \frac{1}{2}\end{array} \right.\).
Tacó:\[A = {x_1}\left( {{x_1} + 2024} \right) + {x_2}\left( {{x_2} + 2025} \right) - {x_2}\]
\[ = x_1^2 + 2024{x_1} + x_2^2 + 2025{x_2} - {x_2}\]
\[ = x_1^2 + 2024{x_1} + x_2^2 + 2024{x_2}\]
\[ = \left( {x_1^2 + 2{x_1}{x_2} + x_2^2} \right) - 2{x_1}{x_2} + \left( {2024{x_1} + 2024{x_2}} \right)\]
\[ = {\left( {{x_1} + {x_2}} \right)^2} - 2{x_1}{x_2} + 2024\left( {{x_1} + {x_2}} \right)\]
\[ = {\left( {\frac{5}{2}} \right)^2} - 2 \cdot \frac{1}{2} + 2024 \cdot \frac{5}{2}\]
\[ = \frac{{25}}{4} - 1 + 5060 = \frac{{20\,\,261}}{4}\].
Vậy \[A = \frac{{20\,\,261}}{4}\].