Question

So sánh

\(\sqrt{19}+\sqrt{21}\) với \(2\sqrt{20}\)

Answer 1

Lời giải:

Xét hiệu

\(\sqrt{19}+\sqrt{21}-2\sqrt{20}=(\sqrt{21}-\sqrt{20})-(\sqrt{20}-\sqrt{19})\)

\(=\frac{1}{\sqrt{21}+\sqrt{20}}-\frac{1}{\sqrt{20}+\sqrt{19}}\)

Dễ thấy \(0< \sqrt{20}+\sqrt{19}< \sqrt{21}+\sqrt{20}\Rightarrow \frac{1}{\sqrt{20}+\sqrt{19}}>\frac{1}{\sqrt{20}+\sqrt{20}}\)

\(\Rightarrow \sqrt{19}+\sqrt{21}-2\sqrt{20}<0\Rightarrow \sqrt{19}+\sqrt{21}< 2\sqrt{20}\)

Answer 2

Cách khác:

\((\sqrt{19}+\sqrt{21})^2=19+21+2\sqrt{19.21}=40+2\sqrt{(20-1)(20+1)}\)

\(=40+2\sqrt{20^2-1}< 40+2\sqrt{20^2}=80\)

\(\Rightarrow \sqrt{19}+\sqrt{21}< \sqrt{80}=2\sqrt{20}\)