\(\Delta=25-4\left(m-4\right)=41-4m\)
a/ Để pt có 2 nghiệm trái dấu
\(\Leftrightarrow ac< 0\Rightarrow m-4< 0\Rightarrow m< 4\)
b/ Để pt có 2 nghiệm dương pb
\(\Leftrightarrow\left\{{}\begin{matrix}\Delta>0\\x_1+x_2>0\\x_1x_2>0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}41-4m>0\\5>0\\m-4>0\end{matrix}\right.\) \(\Rightarrow4< m< \frac{41}{4}\)
c/ Do \(x_1;x_2\) là nghiệm của pt nên:
\(\left\{{}\begin{matrix}x_1^2-5x_1+m-4=0\\x_2^2-5x_2+m-4=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x_1^2-4x_1+m-2=x_1+2\\x_2^2-4x_2+m-2=x_2+2\end{matrix}\right.\)
Thay vào bài toán:
\(x_1\left(x_1+2\right)+x_2\left(x_2+2\right)=23\)
\(\Leftrightarrow x_1^2+x_2^2+2\left(x_1+x_2\right)=23\)
\(\Leftrightarrow\left(x_1+x_2\right)^2-2x_1x_2+2\left(x_1+x_2\right)=23\)
\(\Leftrightarrow25-2\left(m-4\right)-10=23\Rightarrow m=4\)