Discussion

Question:

The expression $2 + \sqrt{2}$ + $\frac{1}{2 + \sqrt{2}}$ + $\frac{1}{\sqrt{2} - 2}$ equals:

Explanation:

$2 + \sqrt{2}$ + $\frac{1}{2 + \sqrt{2}}$ + $\frac{1}{\sqrt{2} - 2}$

= $2 + \sqrt{2}$ + $\frac{1}{2 + \sqrt{2}}$ - $\frac{1}{2 - \sqrt{2}}$

= $\frac{(2 + \sqrt{2}) (2 + \sqrt{2}) (2 - \sqrt{2}) + (2 - \sqrt{2}) - (2 + \sqrt{2})}{(2 + \sqrt{2}) (2 - \sqrt{2})}$

= $\frac{(2 + \sqrt{2}) (4 - 2) + 2 - \sqrt{2} - 2 - \sqrt{2}}{4 - 2}$

= $\frac{4 + 2 \sqrt{2} - 2 \sqrt{2}}{4 - 2}$

= $\frac{4}{2}$

= 2

Hence, option (a).

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