Question

bài 13: tìm tất các giá trị của tham số m để phương trình sau có hai nghiệm dương phân biệt

a) x2+2(m-1)x+3m-3=0

b)x2+(m-2)x+m-1=0

c) x2+(m-2)x+m+1=0

d)-x2-(m-3)x+m+1=0

e)4x2+2(m-1)x+m-1=0

f)(m-2)x2-2(m-2)x+1=0

Answer 1

Để pt có 2 nghiệm dương pb \(\left\{{}\begin{matrix}a\ne0\\\Delta>0\\x_1+x_2=-\frac{b}{a}>0\\x_1x_2=\frac{c}{a}>0\end{matrix}\right.\)

a/ \(\left\{{}\begin{matrix}\Delta'=\left(m-1\right)^2-\left(3m-3\right)>0\\x_1+x_2=-2\left(m-1\right)>0\\x_1x_2=3\left(m-1\right)>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(m-1\right)\left(m-4\right)>0\\m< 1\\m>1\end{matrix}\right.\)

\(\Rightarrow\) Không tồn tại m thỏa mãn

b/ \(\left\{{}\begin{matrix}\Delta=\left(m-2\right)^2-4\left(m-1\right)>0\\x_1+x_2=2-m>0\\x_1x_2=m-1>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m^2-8m+8>0\\m< 2\\m>1\end{matrix}\right.\)

\(\Rightarrow1< m< 4-2\sqrt{2}\)

Answer 2

c/

\(\left\{{}\begin{matrix}\Delta=\left(m-2\right)^2-4\left(m+1\right)>0\\x_1+x_2=2-m>0\\x_1x_2=m+1>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m^2-8m>0\\m< 2\\m>-1\end{matrix}\right.\)

\(\Rightarrow-1< m< 0\)

d/

\(\left\{{}\begin{matrix}\Delta=\left(m-3\right)^2+4\left(m+1\right)>0\\x_1+x_2=3-m>0\\x_1x_2=-m-1>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m^2-2m+13>0\\m< 3\\m< -1\end{matrix}\right.\)

\(\Rightarrow m< -1\)

Answer 3

e/

\(\left\{{}\begin{matrix}\Delta'=\left(m-1\right)^2-4\left(m-1\right)>0\\x_1+x_2=\frac{1-m}{2}>0\\x_1x_2=\frac{m-1}{4}>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(m-1\right)\left(m-5\right)>0\\m< 1\\m>1\end{matrix}\right.\)

Không tồn tại m thỏa mãn

f/

\(\left\{{}\begin{matrix}m-2\ne0\\\Delta'=\left(m-2\right)^2-\left(m-2\right)>0\\x_1+x_2=2>0\\x_1x_2=\frac{1}{m-2}>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m\ne2\\\left(m-2\right)\left(m-3\right)>0\\m-2>0\end{matrix}\right.\)

\(\Rightarrow m>3\)