Question

`x^2 -2(m-1)x-m^2 -3=0`

Tìm m để pt có 2 nghiệm phân biệt x1, x2 (x1 < x2) thỏa mãn \(\left|x_2+3\right|-\left|x_1-2\right|=m+3\)

Answer 1

\(ac=-m^2-3< 0;\forall m\) nên pt luôn có 2 nghiệm pb trái dấu

\(\Rightarrow x_1< 0< x_2\)

\(\Rightarrow\left\{{}\begin{matrix}x_1-2< 0\\x_2+3>0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\left|x_2+3\right|=x_2+3\\\left|x_1-2\right|=2-x_1\end{matrix}\right.\)

Do đó:

\(\left|x_2+3\right|-\left|x_1-2\right|=m+3\)

\(\Leftrightarrow x_2+3-\left(2-x_1\right)=m+3\)

\(\Leftrightarrow x_1+x_2+1=m+3\)

\(\Leftrightarrow2\left(m-1\right)+1=m+3\)

\(\Leftrightarrow m=4\)