\(\frac{y}{2}+\frac{3}{4}\sqrt{1-4y+4y^2}-\frac{3}{2}\)
\(=\frac{y}{2}+\frac{3}{4}\sqrt{\left(2y-1\right)^2}-\frac{3}{2}\)
\(=\frac{y}{2}+\frac{3}{4}\left|2y-1\right|-\frac{3}{2}\)(*)
Theo giả thiết \(y\le\frac{1}{2}\Leftrightarrow2y-1\le0\)
(*) \(=\frac{y}{2}+\frac{3}{4}\cdot\left(1-2y\right)-\frac{3}{2}\)
\(=\frac{y}{2}+\frac{3}{4}-\frac{3y}{2}-\frac{3}{2}\)
\(=\frac{-2y}{2}-\frac{3}{4}\)
\(=-y-\frac{3}{4}\)
\(\sqrt{2+\sqrt{5+4\sqrt{2+\sqrt{9+\sqrt{32}}}}}\)
\(=\sqrt{2+\sqrt{5+4\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}}}\)
\(=\sqrt{2+\sqrt{5+4\sqrt{3+2\sqrt{2}}}}\)
\(=\sqrt{2+\sqrt{5+4\sqrt{\left(\sqrt{2}+1\right)^2}}}\)
\(=\sqrt{2+\sqrt{9+4\sqrt{2}}}\)
\(=\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}\)
\(=\sqrt{3+2\sqrt{2}}=\sqrt{\left(\sqrt{2}+1\right)^2}=\sqrt{2}+1\)
