Let S={x∈R :x≥0 & 2∣∣x√−3∣∣+x√ (x√−6)+6=0} . Then S:Contains exactly four elementsIs an empty setContains exactly one elementContains exactly two elements
Question
Let S={x∈R :x≥0 & 2∣∣x√−3∣∣+x√ (x√−6)+6=0} . Then S:
- Contains exactly four elements
- Is an empty set
- Contains exactly one element
- Contains exactly two elements
Solution
The given equation is a quadratic equation in terms of √x. Let's solve it step by step:
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Let y = √x. Then the equation becomes: 2|y√-3| + y√(y√-6) + 6 = 0.
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This equation is not defined for real numbers because of the terms √-3 and √-6, which are imaginary numbers.
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Therefore, there are no real solutions to this equation.
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Since the set S is defined as S={x∈R :x≥0}, and there are no real solutions to the equation, S is an empty set.
So, the correct answer is: S is an empty set.
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