Question : If $x = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}$ , then what is the value of $(x^{2} + x - 9)$ ?
Option 1: 0
Option 2: $3 \sqrt{2}$
Option 3: $3 \sqrt{3}$
Option 4: $5 \sqrt{3}$

Question

Question : If $x = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}$ and $y = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$ , then the value of $\frac{x^{2} + x y + y^{2}}{x^{2} - x y + y^{2}}$ is:

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Answer 1

Correct Answer: $5 \sqrt{3}$

Solution : Given: $x = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}$
Using the rationalisation method, we get,
⇒ $x = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}}}$
⇒ $x = \sqrt{\frac{(2 + \sqrt{3})^{2}}{2^{2} - {\sqrt{3}}^{2}}}$
⇒ $x = \sqrt{\frac{(2 + \sqrt{3})^{2}}{4 - 3}}$
⇒ $x = (2 + \sqrt{3})$
Putting $x = (2 + \sqrt{3})$ in $x^{2} + x - 9$ , we get,
= $(2 + \sqrt{3})^{2} + (2 + \sqrt{3}) - 9$
= $(4 + 3 + 4 \sqrt{3}) + (2 + \sqrt{3}) - 9$
= $5 \sqrt{3}$
Hence, the correct answer is $5 \sqrt{3}$ .

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Question : If x=2+32−3, then what is the value of (x2+x−9)?Option 1: 0Option 2: 32Option 3: 33Option 4: 53

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Question : If $x = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}$ , then what is the value of $(x^{2} + x - 9)$ ?
Option 1: 0
Option 2: $3 \sqrt{2}$
Option 3: $3 \sqrt{3}$
Option 4: $5 \sqrt{3}$