Question

tìm m để các pt bậc 2 ẩn x sau: \(x^2-2\left(m-1\right)x+m+2=0\) có 2 nghiệm x1,x2 t/m \(\frac{x_1}{x_2}+\frac{x_2}{x_1}=4\)

Answer 1

Tự tìm delta nhé.

Áp dụng Viete: \(\left\{{}\begin{matrix}x_1+x_2=2\left(m-1\right)\\x_1x_2=m+2\end{matrix}\right.\)

\(\frac{x_1}{x_2}+\frac{x_2}{x_1}=\frac{x_1^2+x_2^2}{x_1x_2}=\frac{\left(x_1+x_2\right)^2-2x_1x_2}{x_1x_2}=\frac{\left(2m-2\right)^2-2\left(m+2\right)}{m+2}=4\)

\(\Leftrightarrow4m^2-10m-4m-8=0\)

\(\Leftrightarrow4m^2-14m-8=0\)

\(\Leftrightarrow\left(m-4\right)\left(2m+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}m=4\\m=\frac{-1}{2}\end{matrix}\right.\)