\(\Delta'=\left(m+1\right)^2-\left(m^2+m-1\right)\ge0\)
\(\Leftrightarrow m+2\ge0\Rightarrow m\ge-2\)
Khi đó theo hệ thức Viet : \(\left\{{}\begin{matrix}x_1+x_2=2\left(m+1\right)\\x_1x_2=m^2+m-1\end{matrix}\right.\)
\(x_1^2+x_2^2=\left(x_1+x_2\right)^2-2x_1x_2=4\left(m+1\right)^2-2\left(m^2+m-1\right)=2m^2+6m+6\)
x2 - 2(m + 1)x + m2 + m - 1 = 0
\(\Delta\) = [-2(m + 1)]2 - 4.1.(m2 + m - 1) = 4(m2 + 2m + 1) - 4m2 - 4m + 4 = 4m2 + 8m + 4 - 4m2 - 4m + 4 = 4m + 8
Để pt có nghiệm thì \(\Delta\) \(\ge\) 0 \(\Leftrightarrow\) 4m + 8 \(\ge\) 0 \(\Leftrightarrow\) m \(\ge\) -2
Với m \(\ge\) -2 ta có:
x1 = \(\dfrac{2\left(m+1\right)+\sqrt{4m+8}}{2}=m+1+\sqrt{m+2}\)
x2 = \(\dfrac{2\left(m+1\right)-\sqrt{4m+8}}{2}=m+1-\sqrt{m+2}\)
x1 + x2 = m + 1 + \(\sqrt{m+2}\) + m + 1 - \(\sqrt{m+2}\) = 2m + 2
x1x2 = (m + 1 + \(\sqrt{m+2}\))(m + 1 - \(\sqrt{m+2}\)) = (m + 1)2 - m - 2 = m2 + 2m + 1 - m - 2 = m2 + m - 1 = \(\left(m+\dfrac{1-\sqrt{5}}{2}\right)\left(m+\dfrac{1+\sqrt{5}}{2}\right)\)
(x1)2 + (x2)2 = (m + 1 + \(\sqrt{m+2}\))2 + (m + 1 - \(\sqrt{m+2}\))2 = (x1 + x2)2 - 2x1x2 = (2m + 2)2 - 2(m2 + m - 1) = 4m2 + 8m + 4 - 2m2 - 2m + 2 = 2m2 + 6m + 6 = 2(m2 + 3m + 3)
Chúc bn học tốt!