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Let S={ $x$ $\epsilon$ R : $x\geq$ 0 and

$2\left | \sqrt{x}-3 \right |+{\sqrt{x}}\left ( \sqrt{x}-6 \right )+6=0$

Then S:

Option 1)
contains exactly four elements

Option 2)
is an empty set.

Option 3)
contains exactly one element

Option 4)
contains exactly two elements

Answers (1)

best_answer

Case 1

$\sqrt x \geq 3\Rightarrow x\geq 9$

$2(t-3)+t(t-6)+6=0$

$t^{2}-4t= 0$

$\Rightarrow t= 0 , t= 4$

$\sqrt x= 0 , \sqrt x=4$

$x= 0 , x=16$

we take x= 16 $x\geq 9$

case 2

$0< \sqrt x < 3 \Rightarrow 0< x < 9$

$-2t+6+t^{2}-6t+6= 0$

$t^{2}-8t+12=0$

$\Rightarrow t=2 , t=6$

$\Rightarrow x=4 , x=36$

Thus x= 4 : $x< 9$

So there are two elements

Option 1)

contains exactly four elements

Option 2)

is an empty set.

Option 3)

contains exactly one element

Option 4)

contains exactly two elements

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