Question

Hãy biến đổi: \(2\sqrt{bc}=2\sqrt{ab}+2\sqrt{ca}\Leftrightarrow\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}\)

Answer 1

ta có \(2\sqrt{bc}=2\sqrt{ab}+2\sqrt{ca}\)

\(\Leftrightarrow\sqrt{bc}=\sqrt{ab}+\sqrt{ac}\)

\(\Leftrightarrow\sqrt{bc}=\sqrt{a}\left(\sqrt{b}+\sqrt{c}\right)\)

\(\Leftrightarrow\frac{1}{\sqrt{a}}=\frac{\sqrt{b}+\sqrt{c}}{\sqrt{bc}}\)

\(\Leftrightarrow\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}\)

Answer 2

Ta có : \(2\sqrt{bc}=2\sqrt{ab}+2\sqrt{ca}\)

=> \(\frac{2\sqrt{abc}}{\sqrt{a}}=\frac{2\sqrt{abc}}{\sqrt{c}}+\frac{2\sqrt{abc}}{\sqrt{b}}\)

=> \(2\sqrt{abc}\left(\frac{1}{\sqrt{a}}\right)=2\sqrt{abc}\left(\frac{1}{\sqrt{c}}+\frac{1}{\sqrt{b}}\right)\)

=> \(\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{c}}+\frac{1}{\sqrt{b}}\)