Question

giúp mình nha!!

Answer 1

*Tạm thời mình chỉ làm bài 1 nhé, ngày mai mình làm nốt 2 bài còn lại giúp bn cho.

Để phương trình \(x^2-mx+m^2-4=0\) có hai nghiệm trái dấu thì:

\(\left\{{}\begin{matrix}\Delta>0\\m^2-4< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m^2-4.1.\left(m^2-4\right)>0\\\left(m-2\right)\left(m+2\right)< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-3m^2+16>0\\\left\{{}\begin{matrix}\left[{}\begin{matrix}m-2>0\\m+2< 0\end{matrix}\right.\\\left[{}\begin{matrix}m-2< 0\\m+2>0\end{matrix}\right.\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}3m^2-16< 0\\-2< m< 2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(m\sqrt{3}-4\right)\left(m\sqrt{3}+4\right)< 0\\-2< m< 2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left\{{}\begin{matrix}\left[{}\begin{matrix}m\sqrt{3}-4>0\\m\sqrt{3}+4< 0\end{matrix}\right.\\\left[{}\begin{matrix}m\sqrt{3}-4< 0\\m\sqrt{3}+4>0\end{matrix}\right.\end{matrix}\right.\\-2< m< 2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-\dfrac{4\sqrt{3}}{3}< m< \dfrac{4\sqrt{3}}{3}\\-2< m< 2\end{matrix}\right.\)

\(\Leftrightarrow-2< m< 2\)

Answer 2

2. Để phương trình \(x^2-2mx-6m-9=0\) có hai nghiệm phân biệt âm thì:

\(\left\{{}\begin{matrix}\Delta>0\\2m< 0\\-6m-9>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}4m^2-4.1.\left(-6m-9\right)>0\\m< 0\\m< -\dfrac{3}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}4\left(m^2+6m+9\right)>0\\m< 0\\m< -\dfrac{3}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}4\left(m+3\right)^2>0\\m< 0\\m< -\dfrac{3}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m+3\ne0\\m< 0\\m< -\dfrac{3}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m\ne-3\\m< -\dfrac{3}{2}\end{matrix}\right.\)

Answer 3

3. Để tồn tại hình chữ nói trên thì:

\(\left\{{}\begin{matrix}\Delta\ge0\\2\left(m-3\right)>0\\m-4>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}4\left(m-3\right)^2-4.1.\left(m-4\right)\ge0\\m>3\\m>4\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(m-3\right)^2-\left(m-4\right)\ge0\\m>3\\m>4\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m^2-6m+9-m+4\ge0\\m>3\\m>4\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m^2-7m+13\ge0\\m>3\\m>4\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(m^2-7m+\dfrac{49}{4}\right)+\dfrac{3}{4}\ge0\\m>3\\m>4\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(m-\dfrac{7}{2}\right)^2+\dfrac{3}{4}\ge0\left(BPTVSN\right)\\m>3\\m>4\end{matrix}\right.\)

\(\Leftrightarrow m>4\)