Question

Bài 8: Tìm m để các bất phương trình sau nghiệm đúng với mọi x:

a. \(3x^2+2\left(m-1\right)x+m+4>0\)

b. \(x^2+\left(m+1\right)x+2m+7>0\)

c. \(2x^2+\left(m-2\right)x-m+4>0\)

d. \(mx^2+\left(m-1\right)x+m-1< 0\)

Answer 1

a/ \(\Delta'=\left(m-1\right)^2-3\left(m+4\right)< 0\)

\(\Leftrightarrow m^2-5m-11< 0\Leftrightarrow\frac{5-\sqrt{69}}{2}< m< \frac{5+\sqrt{69}}{2}\)

b/ \(\Delta=\left(m+1\right)^2-4\left(2m+7\right)< 0\)

\(\Leftrightarrow m^2-6m-27< 0\Rightarrow-3< m< 9\)

c/ \(\Delta=\left(m-2\right)^2-8\left(-m+4\right)< 0\)

\(\Leftrightarrow m^2+4m-28< 0\Rightarrow-2-4\sqrt{2}< m< -2+4\sqrt{2}\)

d/ \(\left\{{}\begin{matrix}m< 0\\\Delta=\left(m-1\right)^2-4m\left(m-1\right)< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m< 0\\\left(m-1\right)\left(-3m-1\right)< 0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}m< 0\\\left[{}\begin{matrix}m< -\frac{1}{3}\\m>1\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow m< -\frac{1}{3}\)