8: \(\Delta=\left[-2\left(m+1\right)\right]^2-4\cdot1\left(2m-13\right)\)
\(=4m^2+8m+4-8m+52=4m^2+56>0\forall m\)
=>Phương trình luôn có hai nghiệm phân biệt
Theo Vi-et, ta có:
\(\left\{{}\begin{matrix}x_1+x_2=-\dfrac{b}{a}=2\left(m+1\right)=2m+2\\x_1x_2=\dfrac{c}{a}=2m-13\end{matrix}\right.\)
\(\left(\sqrt{x_1}+\sqrt{x_2}\right)^2=x_1+x_2+2\sqrt{x_1x_2}=2m+2+2\sqrt{2m-13}\)
=>\(\sqrt{x_1}+\sqrt{x_2}=\sqrt{2m+2+2\sqrt{2m-13}}\)
\(C=x_1\sqrt{x_2}+x_2\sqrt{x_1}\)
\(=\sqrt{x_1x_2}\left(\sqrt{x_1}+\sqrt{x_2}\right)\)
\(=\sqrt{2m-13}\cdot\sqrt{2m+2+2\sqrt{2m-13}}\)
\(=\sqrt{\left(2m-13\right)\left(2m+2+2\sqrt{2m-13}\right)}\)
9: \(D=-x_1^2-x_2^2\)
\(=-\left(x_1^2+x_2^2\right)\)
\(=-\left[\left(x_1+x_2\right)^2-2x_1x_2\right]\)
\(=-\left[\left(2m+2\right)^2-2\left(2m-13\right)\right]\)
\(=-\left[4m^2+8m+4-4m+26\right]\)
\(=-\left[4m^2+4m+30\right]\)
\(=-\left(4m^2+4m+1\right)-29=-\left(2m+1\right)^2-29< =-29\forall m\)
Dấu '=' xảy ra khi 2m+1=0
=>2m=-1
=>\(m=-\dfrac{1}{2}\)
10: \(E=x_1+x_2-x_1x_2\)
=2m+2-(2m-13)
=15
=>E không phụ thuộc vào m
