Question

Tìm m để phương trình x² - 2(m+1)x + m² -1= 0 có hai nghiệm x₁, x₂  thỏa \(\dfrac{1}{x_1}+\dfrac{1}{x_2}=\dfrac{1}{6}\)

Answer 1

PT có 2 nghiệm \(\Leftrightarrow\Delta'\ge0\)

\(\Leftrightarrow\left(m+1\right)^2-\left(m^2-1\right)\ge0\\ \Leftrightarrow m^2+2m+1-m^2+1\ge0\\ \Leftrightarrow m\ge-1\)

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

Ta có \(\dfrac{1}{x_1}+\dfrac{1}{x_2}=\dfrac{1}{6}\Leftrightarrow\dfrac{x_1+x_2}{x_1x_2}=\dfrac{1}{6}\)

\(\Leftrightarrow\dfrac{2\left(m+1\right)}{m^2-1}=\dfrac{1}{6}\Leftrightarrow12m+12=m^2-1\\ \Leftrightarrow m^2-12m-13=0\\ \Leftrightarrow\left[{}\begin{matrix}m=13\left(tm\right)\\m=-1\left(tm\right)\end{matrix}\right.\)