Question

cho A=\(\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}}\)

chứng minh:\(A^2-2A-2=0\)

Answer 1

\(A=\sqrt{3+\sqrt{5+2\sqrt{3}}}+\sqrt{3-\sqrt{5+2\sqrt{3}}.}\)

\(\Rightarrow A^2=6+2\sqrt{\left(3+\sqrt{5+2\sqrt{3}}\right)\left(3-\sqrt{5+2\sqrt{3}}\right)}=6+2\sqrt{4-2\sqrt{3}}\)

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

\(\Rightarrow A^2-2A-2=4+2\sqrt{3}-2\left(1+\sqrt{3}\right)-2=0\)

Answer 2

thanks

Answer 3

ket ban di

Answer 4

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

\(\Rightarrow A^2=6+2\sqrt{\left(3+\sqrt{5+2\sqrt{3}}\right)\left(3-\sqrt{5+2\sqrt{3}}\right)}\)\(=6+2\sqrt{4-2\sqrt{3}}\)

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

\(\Rightarrow A^2-2A-2=4+2\sqrt{3}-2\left(1+\sqrt{3}\right)-2=0\)