Thiếu đề rồi bạn ơi, bổ sung nhé:
Cho \(x+y+z+\sqrt{xyz}=4\)
Tính: \(M=\sqrt{x\left(4-y\right)\left(4-z\right)}+\sqrt{y\left(4-z\right)\left(4-x\right)}+\sqrt{z\left(4-x\right)\left(4-y\right)}-\sqrt{xyz}\)
Bài làm:
Ta có: \(\sqrt{x\left(4-x\right)\left(4-z\right)}=\sqrt{x\left(16-4y-4z+yz\right)}\)
\(=\sqrt{x\left[4\left(4-y-z\right)+yz\right]}=\sqrt{x\left[4\left(x+\sqrt{xyz}\right)+yz\right]}\)
\(=\sqrt{x\left(4x+4\sqrt{xyz}+yz\right)}=\sqrt{x\left(2\sqrt{x}+\sqrt{yz}\right)^2}\)
\(=\sqrt{\left(2x+\sqrt{xyz}\right)^2}=2x+\sqrt{xyz}\)
Tương tự ta chứng minh được:
\(\sqrt{y\left(4-z\right)\left(4-x\right)}=2y+\sqrt{xyz}\) ; \(\sqrt{z\left(4-x\right)\left(4-y\right)}=2z+\sqrt{xyz}\)
=> \(M=2x+\sqrt{xyz}+2y+\sqrt{xyz}+2z+\sqrt{xyz}-\sqrt{xyz}\)
\(M=2\left(x+y+z+\sqrt{xyz}\right)=2.4=8\)
