ĐKXĐ:\(x-1\ge0\Leftrightarrow x\ge1\)
\(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}=\dfrac{3+x}{2}\)
\(\Leftrightarrow\left(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\right)^2=\left(\dfrac{3+x}{2}\right)^2\)
\(\Leftrightarrow\left(\sqrt{x+2\sqrt{x-1}}\right)^2+2.\sqrt{x+2\sqrt{x-1}}.\sqrt{x-2\sqrt{x-1}}+\left(\sqrt{x-2\sqrt{x-1}}\right)^2=\dfrac{x^2+6x+9}{4}\)
\(\Leftrightarrow x+2\sqrt{x-1}+2\sqrt{\left(x+2\sqrt{x-1}\right)\left(x-2\sqrt{x-1}\right)}+x-2\sqrt{x-1}=\dfrac{x^2+6x+9}{4}\)\(\Leftrightarrow2x+2\sqrt{x^2-\left(2\sqrt{x-1}\right)^2}=\dfrac{x^2+6x+9}{4}\)
\(\Leftrightarrow2x+2\sqrt{x^2-\left(2^2.\sqrt{x-1}^2\right)}=\dfrac{x^2+6x+9}{4}\)
\(\Leftrightarrow2x+2\sqrt{x^2-4x+4}=\dfrac{x^2+6x+9}{4}\)
\(\Leftrightarrow2x+2\sqrt{\left(x-2\right)^2}=\dfrac{x^2+6x+9}{4}\)
\(\Leftrightarrow2x+2.\left(x-2\right)=\dfrac{x^2+6x+9}{4}\)
\(\Leftrightarrow2x+2x+4=\dfrac{x^2+6x+9}{4}\)
\(\Leftrightarrow4x+4=\dfrac{x^2+6x+9}{4}\)
\(\Leftrightarrow4\left(4x-4\right)=x^2+6x+9\)
\(\Leftrightarrow16x-16=x^2+6x+9\)
\(\Leftrightarrow16x-x^2-6x=16+9\)
\(\Leftrightarrow10x-x^2=25\)
\(\Leftrightarrow10x-x^2-25=0\)
\(\Leftrightarrow-\left(x^2-10x+25\right)=0\)
\(\Leftrightarrow-\left(x-5\right)^2=0\)
\(\Leftrightarrow-\left(x-5\right)=0\)
\(\Leftrightarrow-x+5=0\)
\(\Leftrightarrow-x=-5\)
\(\Leftrightarrow x=5\) (TMĐKXĐ)
