# The sum of the solutions of the equation $\left | \sqrt{x}-2 \right |+\sqrt{x}\left (\sqrt{x}-4 \right )+2=0,(x>0)$ is equal to : Option 1) $12$ Option 2) $10$ Option 3) $9$ Option 4) $4$

$\left | \sqrt{x}-2 \right |+\sqrt{x}\left (\sqrt{x}-4 \right )+2=0,(x>0)$

$\\\left | \sqrt{x}-2 \right |+x-4\sqrt{x}+2=0\\\; \; \; \\\left | \sqrt{x}-2 \right |+\left ( \sqrt{x}-2 \right )^{2}-4+2=0$

$\left | \sqrt{x}-2 \right |+\left ( \sqrt{x}-2 \right )^{2}-2=0$

$Let\; \left | \sqrt{x}-2 \right |=t$

$t+t^{2}-2=0$

$t^{2}+t-2=0$

$t=\frac{-1\pm \sqrt{1+3}}{2}=\frac{-1\pm 3}{2}=-2,1$

$\left | \sqrt{x}-2 \right |=1\; \; \; \; \; \; (as\; t>0)$

$\sqrt{x}-2 =\pm 1$

$\sqrt{x} =+1+2=3,$

$=-1+2=1,$

$\\\sqrt{x}=3 , \; \; \; \; x=9\\\; \; \; \\\sqrt{x}=1 \; \; \; \; x=1$

Sum $=9+1=10$

Option 1)

$12$

Option 2)

$10$

Option 3)

$9$

Option 4)

$4$

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