Question

Cho \(x=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}.\)Tính giá trị của biểu thức A=x²-2x-2

Answer 1

Ta có:

\(x=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}\)   ( x> 0 )

\(\Rightarrow x^2=6+2\sqrt{\left(3+\sqrt{5+2\sqrt{3}}\right)\left(3-\sqrt{5+2\sqrt{3}}\right)}\)

\(=6+2\sqrt{9-5-2\sqrt{3}}\)

\(=6+2\sqrt{\left(\sqrt{3}-1\right)^2}\)

\(=6+2\sqrt{3}-2=4+2\sqrt{3}=\left(\sqrt{3}+1\right)^2\)

\(\Rightarrow x=\sqrt{3}+1\)

Vậy :

\(A=x^2-2x-2=4+2\sqrt{3}-2\sqrt{3}-2-2\)

\(=0\)