Q. Let $S = \{ x \in R : x \geq 0$ and $ 2 | \sqrt{x} - 3 | + \sqrt{x} ( \sqrt{x} - 6 ) + 6 = 0 \}$ . Then S :

Solution:

Case - I : $ x \in [0, 9]$
$ 2(3 - \sqrt{x} ) + x - 6 \sqrt{x} + 6 = 0$
$\Rightarrow \, \, x - 8 \sqrt{x} + 12 = 0 \, \, \Rightarrow \, \, \sqrt{ x} = 4, 2 x = 16 , 4 \Rightarrow \, x = 4 $
Case-II : $x \in [9, \infty]$
$2( \sqrt{x} -3) + x - 6 \sqrt{x} + 6 = 0 \, x - 4 \sqrt{x} = 0 \, \Rightarrow \, x = 16, 0$
So x = 4, 16