ĐK:\(x\ge4\)
\(2\sqrt{x-4}+\sqrt{x+1}=\sqrt{2x-3}+\sqrt{4x-16}\)
\(\Leftrightarrow2\sqrt{x-4}+\sqrt{x+1}-\sqrt{5}=\sqrt{2x-3}-\sqrt{5}+\sqrt{4x-16}\)
\(\Leftrightarrow2\sqrt{x-4}+\frac{x+1-5}{\sqrt{x+1}+\sqrt{5}}=\frac{2x-3-25}{\sqrt{2x-3}+\sqrt{5}}+\sqrt{4\left(x-4\right)}\)
\(\Leftrightarrow2\sqrt{x-4}+\frac{x-4}{\sqrt{x+1}+\sqrt{5}}-\frac{2\left(x-4\right)}{\sqrt{2x-3}+\sqrt{5}}-\sqrt{4\left(x-4\right)}=0\)
\(\Leftrightarrow\left(x-4\right)\left(\frac{2}{\sqrt{x-4}}+\frac{1}{\sqrt{x+1}+\sqrt{5}}-\frac{2}{\sqrt{2x-3}+\sqrt{5}}-\frac{2}{\sqrt{x-4}}\right)=0\)
\(\Leftrightarrow\left(x-4\right)\left(\frac{1}{\sqrt{x+1}+\sqrt{5}}-\frac{2}{\sqrt{2x-3}+\sqrt{5}}\right)=0\)
\(\Rightarrow x=4\). Và \(\sqrt{2x-3}+\sqrt{5}=2\sqrt{x+1}+2\sqrt{5}\)
\(\Leftrightarrow\sqrt{2x-3}=2\sqrt{x+1}+\sqrt{5}\)
\(\Leftrightarrow2x-3=4x+9+4\sqrt{5\left(x+1\right)}\)
\(\Leftrightarrow-2x-12=4\sqrt{5\left(x+1\right)}\)*vô nghiệm vì \(VT< 0;VP>0\forall x\ge4\)*
Vậy \(x=4\)
