ĐK: \(x\in R\)
\(\left(x+\sqrt{x^2+9}\right)\left(y+\sqrt{y^2+9}\right)=9\left(1\right)\)
\(\Leftrightarrow9\left(y+\sqrt{y^2+9}\right)=9\left(\sqrt{x^2+9}-x\right)\)
\(\Leftrightarrow x+y=\sqrt{x^2+9}-\sqrt{y^2+9}\left(3\right)\)
Mặt khác: \(\left(1\right)\Leftrightarrow9\left(x+\sqrt{x^2+9}\right)=9\left(\sqrt{y^2+9}-y\right)\)
\(\Leftrightarrow\sqrt{x^2+9}-\sqrt{y^2+9}=-x-y\left(4\right)\)
Từ \(\left(3\right)\&\left(4\right)\Rightarrow x=-y\)
Khi đó: \(M=x^4+2x^2-8x+2024\)
\(=\left(x^4-2x^2+1\right)+\left(4x^2-8x+4\right)+2019\)
\(=\left(x^2-1\right)^2+4\left(x-1\right)^2+2019\ge2019\)
\(minM=2019\Leftrightarrow\left(x;y\right)=\left(1;-1\right)\)
