\(\dfrac{2\sqrt{2}}{\sqrt{x}+1}+\sqrt{x}=\sqrt{x+9}\)
\(\Leftrightarrow\dfrac{2\sqrt{2}}{\sqrt{x+1}}+\sqrt{x}-\sqrt{x+9}=0\)
\(\Leftrightarrow\dfrac{2\sqrt{2}+\sqrt{\left(x+1\right)\cdot x}-\sqrt{\left(x+1\right)\cdot\left(x+9\right)}}{\sqrt{x+1}}=0\)\(Suyra:2\sqrt{2}+\sqrt{x^2+x}-\sqrt{x^2+10x+9=0}\)(DKXD:\(x\ne-1;x\ne-9\))
\(\Leftrightarrow\sqrt{x^2+x}=-2\sqrt{2}+\sqrt{x^2+10x+9}\)
\(\Leftrightarrow x^2+x=x^2+10x+9-4\sqrt{\left(x^2+10x+9\right)\cdot2}+8\)\(\Leftrightarrow x=10x+9-4\sqrt{2x^2+20x+18}+8\)
\(\Leftrightarrow x=10x+17-4\sqrt{2x^2+20x+18}\)
\(\Leftrightarrow4\sqrt{2x^2+20x+18}=9x+17\)
\(\Leftrightarrow\left(4\sqrt{2x^2+20x+18}\right)^2=\left(9x+17\right)^2\)
\(\Leftrightarrow16\left(2x^2+20+18\right)=81x^2+306x+289\)\(\Leftrightarrow32x^2+320x+288-81x^2-306x-289=0\)\(\Leftrightarrow-49x^2+14x-1=0\)
\(\Leftrightarrow-49x^2+7x+7x-1=0\)
\(\Leftrightarrow-7x\cdot\left(7x-1\right)+\left(7x-1\right)=0\)
\(\Leftrightarrow-\left(7x-1\right)\cdot\left(7x-1\right)=0\)
\(\Leftrightarrow-\left(7x-1\right)^2=0\)
\(\Leftrightarrow7x-1=0\)
\(\Leftrightarrow x=\dfrac{1}{7}\left(TM\right)\)
\(\dfrac{2\sqrt{2}}{\sqrt{x+1}}+\sqrt{x}=\sqrt{x+9}\) (*)\(\)
ĐKXĐ là : x \(\ge0\)
(*) \(\Leftrightarrow\) \(\left(\dfrac{2\sqrt{2}}{\sqrt{x+1}}+\sqrt{x}\right)^2=\left(\sqrt{x+9}\right)^2\)
\(\Leftrightarrow\dfrac{8}{x+1}+\dfrac{4\sqrt{2x}}{\sqrt{x+1}}+x=x+9\)
\(\Leftrightarrow\dfrac{8}{x+1}+\dfrac{4\sqrt{2x\left(x+1\right)}}{x+1}+\dfrac{x\left(x+1\right)}{x+1}=\dfrac{\left(x+1\right)\left(x+9\right)}{x+1}\)
\(\Leftrightarrow\) 8 + 4\(\sqrt{2x^2+2x}\) + x2 + x = x2 + 9x + x + 9
\(\Leftrightarrow\) x2 - x2 + 4\(\sqrt{2x^2+2x}\)= x - x + 9x + 9 - 8
\(\Leftrightarrow4\sqrt{2x^2+2x}\) = 9x + 1
\(\Leftrightarrow\left(4\sqrt{2x^2+2x}\right)^2=\left(9x+1\right)^2\)
\(\Leftrightarrow\) 16 (2x2 + 2x) = 81x2 +18x + 1
\(\Leftrightarrow32x^2+32x-81x^2-18x-1=0\)
\(\Leftrightarrow-49x^2+14x-1=0\)
\(\Leftrightarrow-\left(49x^2-14x+1\right)=0\)
\(\Leftrightarrow-\left(7x-1\right)^2=0\)
\(\Leftrightarrow7x-1=0\)
\(\Leftrightarrow7x=1\)
\(\Leftrightarrow x=\dfrac{1}{7}\left(TMĐK\right)\)
Vậy S = \(\left\{\dfrac{1}{7}\right\}\)
