\(x^2+\left(2m-1\right)x+m+5>=0\)
\(\text{Δ}=\left(2m-1\right)^2-4\cdot1\cdot\left(m+5\right)\)
\(=4m^2-4m+1-4\left(m+5\right)\)
\(=4m^2-8m-19\)
Để BPT này luôn đúng với mọi x thì \(\left\{{}\begin{matrix}\text{Δ}< =0\\a>0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}4m^2-8m-19< =0\\1>0\left(đúng\right)\end{matrix}\right.\)
=>\(4m^2-8m-19< =0\)
=>\(4m^2-8m+4-23< =0\)
=>\(\left(2m-2\right)^2< =23\)
=>\(-\sqrt{23}< =2m-2< =\sqrt{23}\)
=>\(-\sqrt{23}+2< =2m< =\sqrt{23}+2\)
=>\(\dfrac{-\sqrt{23}+2}{2}< =m< =\dfrac{\sqrt{23}+2}{2}\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=1>0\\\Delta=\left(2m-1\right)^2-4\left(m+5\right)\le0\end{matrix}\right.\)
\(\Leftrightarrow4m^2-8m-19\le0\)
\(\Rightarrow\dfrac{2-\sqrt{33}}{2}\le m\le\dfrac{2+\sqrt{33}}{2}\)
