Question

m(m+8)x²+2(m+8)x+9m+1 ≥0 tìm m để bất phương trình vô nghiệm

Answer 1

Xét \(f_{\left(x\right)}=m\left(m+8\right)x^2+2\left(m+8\right)x+9m+1\ge0\)

\(\Leftrightarrow\left(m^2+8m\right).x^2+2\left(m+8\right).x+9m+1\ge0\)

Để bpt vô nghiệm \(\left\{{}\begin{matrix}m^2+8m< 0\\9m^3-72m^2+8m+64< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-3< m< 0\\\left\{{}\begin{matrix}-3< m< \approx\dfrac{-3}{\sqrt{10}}\\m< \approx\dfrac{-3}{\sqrt{10}}\end{matrix}\right.\end{matrix}\right.\)

=> \(-8< m< -\dfrac{3}{\sqrt{10}}\)