Question

Giải phương trình:

\(\sqrt{2x-2+2\sqrt{2x-3}}+\sqrt{2x+13+8\sqrt{2x-3}}=5\)

 

Answer 1

Ta có: \(\sqrt{2x-2+2\sqrt{2x-3}+\sqrt{2x+13+8\sqrt{2x-3}}}=5\)

\(\Leftrightarrow\sqrt{2x-2+2\sqrt{2x-3}+2\sqrt{2x-3}+4}=5\)

\(\Leftrightarrow\sqrt{2x+2+4\sqrt{2x-3}}=5\)

\(\Leftrightarrow\sqrt{2x-3+2\cdot\sqrt{2x-3}\cdot2+4+1}=5\)

\(\Leftrightarrow\left(\sqrt{2x-3}+2\right)^2+1=25\)

\(\Leftrightarrow\left(\sqrt{2x-3}+2\right)^2=24\)

\(\Leftrightarrow\sqrt{2x-3}+2=2\sqrt{6}\)

\(\Leftrightarrow2x-3=\left(2\sqrt{6}-2\right)^2\)

\(\Leftrightarrow2x-3=28-8\sqrt{6}\)

\(\Leftrightarrow2x=31-8\sqrt{6}\)

hay \(x=\dfrac{31-8\sqrt{6}}{2}\)

Answer 2