First slide

Theory of expressions

Question

If $x = \sqrt{7 - 4 \sqrt{3}},$ then $x + \frac{1}{x}$ is equal to:

Moderate

A

2
B

$3 \sqrt{7}$
C

4
D

$4 \sqrt{7}$

Solution

$7 - 4 \sqrt{3} = 2^{2} + (\sqrt{3})^{2} - 4 \sqrt{3}$
$= (2 - \sqrt{3})^{2}$
$∴ x = \sqrt{7 - 4 \sqrt{3}} = 2 - \sqrt{3}$
and $\frac{1}{x} = \frac{1}{2 - \sqrt{3}} = 2 + \sqrt{3}$
Thus, $x + \frac{1}{x} = 4$

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