Question

Cho a, b, c là ba sô thực dương thỏa mãn điều kiện \(a+b+c+\sqrt{abc}=4\). Tính gt biểu thức:

\(A=\sqrt{a\left(4-b\right)\left(4-c\right)}+\sqrt{b\left(4-a\right)\left(4-c\right)}+\sqrt{c\left(4-a\right)\left(4-b\right)}-\sqrt{abc}\)

Giúp với!!

Answer 1

\(a+b+c+\sqrt{abc}=4\Rightarrow4a+4b+4c+4\sqrt{abc}=16\Rightarrow16-4b-4c=4a+4\sqrt{abc}\)

\(\sqrt{a\left(4-b\right)\left(4-c\right)}=\sqrt{a\left(16-4b-4c+bc\right)}=\sqrt{a\left(4a+4\sqrt{abc}+bc\right)}\)

\(=\sqrt{4a^2+4a\sqrt{abc}+abc}=\sqrt{\left(2a+\sqrt{abc}\right)^2}=2a+\sqrt{abc}\)

Tương tự : \(\sqrt{b\left(4-a\right)\left(4-c\right)}=2b+\sqrt{abc}\); \(\sqrt{c\left(4-a\right)\left(4-b\right)}=2c+\sqrt{abc}\)

\(\Rightarrow A=2a+2b+2c+3\sqrt{abc}-\sqrt{abc}=2\left(a+b+c+\sqrt{abc}\right)=8\)