\(x^2-\left(2m+5\right)x-2m-6=0\)
\(\Delta=\left(2m+5\right)^2-4.\left(-2m-6\right)\)
\(=4m^2+20m+25+8m+24\)
\(=4m^2+28m+49\)
\(=\left(2m+7\right)^2\) \(>0\) \(\left(m\ne-\frac{7}{2}\right)\)
Theo hệ thức Vi-ét:
\(\left\{{}\begin{matrix}x_1+x_2=2m+5\\x_1.x_2=-2m-6\end{matrix}\right.\)
Theo bài ra \(\left|x_1\right|+\left|x_2\right|=7\Rightarrow\left(\left|x_1\right|+\left|x_2\right|\right)^2=49\)
\(\Rightarrow\left(x_1+x_2\right)^2-2x_1x_2+2.\left(\left|x_1x_2\right|\right)=49\)
\(\Leftrightarrow\left(2m+5\right)^2-2\left(-2m-6\right)+2.\left(\left|-2m-6\right|\right)=49\)
\(\Leftrightarrow\left(2m+5\right)^2=49\)
\(\Leftrightarrow4m^2+20m+25-49=0\Leftrightarrow4m^2+20m-24=0\)
\(\left\{{}\begin{matrix}m=1\left(l\right)\\m=-6\left(n\right)\end{matrix}\right.\)
đạt giá trị nhỏ nhất