\(\Delta'=\left(m+1\right)^2-\left(2m+10\right)=m^2-9\ge0\Rightarrow\left[{}\begin{matrix}m\ge3\\m\le-3\end{matrix}\right.\)
Theo hệ thức Viet: \(\left\{{}\begin{matrix}x_1+x_2=2\left(m+1\right)\\x_1x_2=2m+10\end{matrix}\right.\)
a. \(\left\{{}\begin{matrix}x_1+x_2=2\left(m+1\right)\\x_1=3x_2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}4x_2=2\left(m+1\right)\\x_1=3x_2\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x_2=\dfrac{m+1}{2}\\x_1=\dfrac{3\left(m+1\right)}{2}\end{matrix}\right.\)
Lại có \(x_1x_2=2m+10\Rightarrow\left(\dfrac{m+1}{2}\right)\left(\dfrac{3\left(m+1\right)}{2}\right)=2m+10\)
\(\Leftrightarrow3m^2+6m+3=8m+40\)
\(\Leftrightarrow3m^2-2m-37=0\Rightarrow m=\dfrac{1\pm4\sqrt{7}}{3}\)
b.
\(P=-\left(x_1+x_2\right)^2-8x_1x_2\)
\(=-4\left(m+1\right)^2-8\left(2m+10\right)\)
\(=-4m^2-24m-84=-4\left(m+3\right)^2-48\le-48\)
\(P_{max}=-48\) khi \(m=-3\)
a) Ta có: \(\Delta=\left[-2\left(m+1\right)\right]^2-4\cdot1\cdot\left(2m+10\right)\)
\(=\left(2m+2\right)^2-4\left(2m+10\right)\)
\(=4m^2+8m+4-8m-40\)
\(=4m^2-36\)
Để phương trình có nghiệm thì \(4m^2-36\ge0\)
\(\Leftrightarrow4m^2\ge36\)
\(\Leftrightarrow m^2\ge9\)
\(\Leftrightarrow\left[{}\begin{matrix}m\ge3\\m\le-3\end{matrix}\right.\)
Khi \(m\ge3\) hoặc \(m\le-3\) thì Áp dụng hệ thức Vi-et, ta được:
\(\left\{{}\begin{matrix}x_1\cdot x_2=2m+10\\x_1+x_2=2\left(m+1\right)=2m+2\end{matrix}\right.\)
mà \(x_1-3x_2=0\) nên ta lập được hệ phương trình:
\(\left\{{}\begin{matrix}x_1+x_2=2m+2\\x_1-3x_2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}4x_2=2m+2\\x_1=3x_2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x_1=3\cdot x_2\\x_2=\dfrac{m+1}{2}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x_1=\dfrac{3m+3}{2}\\x_2=\dfrac{m+1}{2}\end{matrix}\right.\)
Thay \(x_1=\dfrac{3m+3}{2};x_2=\dfrac{m+1}{2}\) vào \(x_1\cdot x_2=2m+10\), ta được:
\(\dfrac{3m+3}{2}\cdot\dfrac{m+1}{2}=2m+10\)
\(\Leftrightarrow\dfrac{3\left(m+1\right)^2}{4}=2m+10\)
\(\Leftrightarrow3\left(m^2+2m+1\right)=8m+40\)
\(\Leftrightarrow3m^2+6m+3-8m-40=0\)
\(\Leftrightarrow3m^2-2m-37=0\)
\(\Delta=\left(-2\right)^2-4\cdot3\cdot\left(-37\right)=4+444=448>0\)
Vì \(\Delta>0\) nên phương trình có hai nghiệm phân biệt là:
\(\left\{{}\begin{matrix}m_1=\dfrac{2+8\sqrt{7}}{6}=\dfrac{4\sqrt{7}+1}{3}\left(nhận\right)\\m_2=\dfrac{2-8\sqrt{7}}{6}=\dfrac{1-4\sqrt{7}}{3}\left(nhận\right)\end{matrix}\right.\)