1: Khi m=2 thì phương trình sẽ trở thành:
\(x^2-2\left(2+1\right)x+2^2+2-1=0\)
=>\(x^2-6x+4+2-1=0\)
=>\(x^2-6x+5=0\)
=>(x-1)(x-5)=0
=>\(\left[\begin{array}{l}x-1=0\\ x-5=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=1\left(nhận\right)\\ x=5\left(nhận\right)\end{array}\right.\)
2: \(\Delta=\left\lbrack2\left(m+1\right)\right\rbrack^2-4\left(m^2+m-1\right)\)
\(=4\left(m+1\right)^2-4\left(m^2+m-1\right)=4\left(m^2+2m+1-m^2-m+1\right)=4\left(m+2\right)\)
Để phương trình có hai nghiệm thì 4(m+2)>=0
=>m+2>=0
=>m>=-2
Ta có: \(\left(\sqrt{x_1^2+1}+x_1\right)\left(\sqrt{x_2^2+1}+x_2\right)=1\)
=>\(\left(\sqrt{x_1^2+1}+x_1\right)\left(\sqrt{x_1^2+1}-x_1\right)\left(\sqrt{x_2^2+1}+x_2\right)=1\cdot\left(\sqrt{x_1^2+1}-x_1\right)\)
=>\(\sqrt{x_2^2+1}+x_2=\sqrt{x_1^2+1}-x_1\)
=>\(\) \(\sqrt{x_2^2+1}-\sqrt{x_1^2+1}=-x_1-x_2\)
=>\(\frac{x_2^2+1-x_1^2-1}{\sqrt{x_2^2+1}+\sqrt{x_1^2+1}}+\left(x_2+x_1\right)=0\)
=>\(\left(x_2+x_1\right)\cdot\frac{\left(x_2-x_1\right)}{\sqrt{x_2^2+1}+\sqrt{x_1^2+1}}+\left(x_2+x_1\right)=0\)
=>\(\left(x_2+x_1\right)\cdot\left\lbrack\frac{\left(x_2-x_1\right)}{\sqrt{x_2^2+1}+\sqrt{x_1^2+1}}+1\right\rbrack=0\)
=>\(x_2+x_1=0\)
=>2(m+1)=0
=>m=-1(nhận)
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