Question

Cho pt : x^2-2(m+1)× x + m^2 + 2 = 0 M? Để pt có p biệt x1 ,x2 t/ mãn |x1 - x2|= 2

Answer 1

\(\Delta'=\left(m+1\right)^2-\left(m^2+2\right)=2m-1>0\Rightarrow m>\dfrac{1}{2}\)

Theo hệ thức Viet: \(\left\{{}\begin{matrix}x_1+x_2=2\left(m+1\right)\\x_1x_2=m^2+2\end{matrix}\right.\)

\(\left|x_1-x_2\right|=2\)

\(\Leftrightarrow\left(x_1-x_2\right)^2=4\)

\(\Leftrightarrow\left(x_1+x_2\right)^2-4x_1x_2=4\)

\(\Leftrightarrow4\left(m+1\right)^2-4\left(m^2+2\right)=4\)

\(\Leftrightarrow8m-4=4\)

\(\Rightarrow m=1\) (thỏa mãn)