Question - Solving Expression with Reciprocal
Solution:
To solve the expression x + $$\frac{1}{x}$$ given that $$x = 2 + \sqrt{3}$$, first find the reciprocal of x and then add it to x.Given $$x = 2 + \sqrt{3}$$, the reciprocal, $$\frac{1}{x}$$, can be calculated as follows:$$\frac{1}{x} = \frac{1}{2 + \sqrt{3}}$$To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:$$\frac{1}{x} = \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})}$$Now, apply the difference of squares to the denominator:$$\frac{1}{x} = \frac{2 - \sqrt{3}}{4 - (\sqrt{3})^2} = \frac{2 - \sqrt{3}}{4 - 3} = \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3}$$Now, add x to $$\frac{1}{x}$$:$$x + \frac{1}{x} = (2 + \sqrt{3}) + (2 - \sqrt{3})$$When you combine the terms, the $$\sqrt{3}$$ terms will cancel out:$$x + \frac{1}{x} = 2 + \sqrt{3} + 2 - \sqrt{3} = 2 + 2 = 4$$So, $$x + \frac{1}{x}$$ is equal to 4.
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