`a)` Ptr có:`\Delta' =[-(m-1)]^2-(-3-m)`
`=m^2-2m+1+3+2m=m^2+4 > 0 AA m`
`=>` Ptr có `2` nghiệm `AA m`
`b) AA m`, áp dụng Vi-ét có:`{(x_1+x_2=[-b]/a=2m-2),(x_1.x_2=c/a=-3-m):}`
Ta có:`x_1 ^2+x_2 ^2 >= 10`
`<=>(x_1+x_2)^2-2x_1.x_2 >= 10`
`<=>(2m-2)^2-2(-3-m) >= 10`
`<=>4m^2-8m+4+6+2m >= 10`
`<=>4m^2-6m+10 >= 10`
`<=>4m^2-6m >= 0`
`<=>2m(2m-3) >= 0`
`<=>` $\left[\begin{matrix} m \ge \dfrac{3}{2}\\ m \le 0\end{matrix}\right.$
Vậy `m >= 3/2` hoặc `m <= 0` thì t/m yêu cầu đề bài
a: \(\text{Δ}=\left(2m-2\right)^2-4\left(-m-3\right)\)
\(=4m^2-8m+4+4m+12=4m^2-4m+16\)
\(=\left(2m-1\right)^2+15>0\)
Do đó: Phương trình luôn có hai nghiệm
b: Theo Vi-et, ta được:
\(\left\{{}\begin{matrix}x_1+x_2=2m-2\\x_1x_2=-m-3\end{matrix}\right.\)
Ta có: \(x_1^2+x_2^2>=10\)
\(\Leftrightarrow\left(x_1+x_2\right)^2-2x_1x_2>=10\)
\(\Leftrightarrow\left(2m-2\right)^2-2\left(-m-3\right)>=10\)
\(\Leftrightarrow4m^2-8m+4+2m+6-10>=0\)
\(\Leftrightarrow4m^2-6m>=0\)
=>2m(2m-3)>=0
=>m>=3/2 hoặc m<=0
a, Ta có:
\(\Delta'=\left[-\left(m-1\right)\right]^2-1\left(-3-m\right)\\ =\left(m-1\right)^2-\left(-3-m\right)\\ =m^2-2m+1+3+m\\ =m^2-m+4\\ =\left(m^2-m+\dfrac{1}{4}\right)+\dfrac{15}{4}\\ =\left(m-\dfrac{1}{2}\right)^2+\dfrac{15}{4}>0\)
Suy ra pt luôn có 2 nghiệm phân biệt
b, Theo Vi-ét:\(x_1+x_2=2m-2;x_1x_2=-m-3\)
\(x_1^2+x_2^2\ge10\\ \Leftrightarrow\left(x_1+x_2\right)^2-2x_1x_2\ge10\\ \Leftrightarrow\left(2m-2\right)^2-2\left(-m-3\right)-10\ge0\\ \Leftrightarrow4m^2-8m+4+2m+6-10\ge0\\ \Leftrightarrow4m^2-6m\ge0\\ \Leftrightarrow2m\left(2m-3\right)\ge0\\ \Leftrightarrow m\left(2m-3\right)\ge0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}m\ge0\\2m-3\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}m\le0\\2m-3\le0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}m\ge0\\m\ge\dfrac{3}{2}\end{matrix}\right.\\\left\{{}\begin{matrix}m\le0\\m\le\dfrac{3}{2}\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}m\ge\dfrac{3}{2}\\m\le0\end{matrix}\right.\)