Theory of Equations 1 Question 12
13. Let $S={x \in \mathbf{R}: x \geq 0$ and $2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0$. Then, $S$
(2018 Main)
(a) is an empty set
(b) contains exactly one element
(c) contains exactly two elements
(d) contains exactly four elements
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Solution:
- We have, $2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0$
$$ \begin{aligned} & \text { Let } \sqrt{x}-3=y \\ & \Rightarrow \quad \sqrt{x}=y+3 \\ & \therefore \quad 2|y|+(y+3)(y-3)+6=0 \\ & \Rightarrow \quad 2|y|+y^{2}-3=0 \\ & \Rightarrow \quad|y|^{2}+2|y|-3=0 \\ & \Rightarrow \quad(|y|+3)(|y|-1)=0 \\ & \Rightarrow \quad|y| \neq-3 \Rightarrow|y|=1 \\ & \Rightarrow \quad y= \pm 1 \Rightarrow \sqrt{x}-3= \pm 1 \\ & \Rightarrow \quad \sqrt{x}=4,2 \Rightarrow x=16,4 \end{aligned} $$
