Let S = \ x ∈ R: x ≥ 0 and 2| x - 3| + x ( x - 6) + 6 = 0\ . Then S...

Let

S = \left\{x \in R: x \ge 0 \text{ and } 2\left|\sqrt{x} - 3\right| + \sqrt{x}\left(\sqrt{x} - 6\right) + 6 = 0\right\}

. Then

S

: $Let S = \ x ∈ R: x ≥ 0 and 2| x - 3| + x ( x - 6) + 6 = 0\ . Then S :$

Options

AContains exactly four elements
BIs an empty set
CContains exactly one element
DContains exactly two elements

Correct answer

D. Contains exactly two elements

Step-by-step solution

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

Let $\sqrt{x} = t > 0$

$2\left|t - 3\right| + t\left(t - 6\right) + 6 = 0$

$⇒ 2\left|t - 3\right|$ $= - \left({t}^{2} - 6t + 6\right)$

$⇒ 2\left|t - 3\right| = - \left({t}^{2} - 6t + 9\right) + 3$
$Let S = \ x ∈ R: x ≥ 0 and 2| x - 3| + x ( x - 6) + 6 = 0\ . Then S :$

So the equation $2$ solutions.

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