Question

If x=3+2√2,check whether
$x + \frac{1}{x}$
is a rational or irrational.

​

Answer 1

$x = 3 + 2 \sqrt{2}$

$\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} }$

$\frac{1}{x} = \frac{1}{ \sqrt{9} + \sqrt{8} }$

$\frac{1}{x} = \frac{1}{ \sqrt{9} + \sqrt{8} } \times \frac{ \sqrt{9} - \sqrt{8} }{ \sqrt{9} - \sqrt{8} }$

$\frac{1}{x} = \frac{ \sqrt{9} - \sqrt{8} }{9 - 8}$

$\frac{1}{x} = 3 - 2 \sqrt{2}$

$x + \frac{1}{x} = 3 + 2 \sqrt{2} + 3 - 2 \sqrt{2}$

$x + \frac{1}{x} = 6$

$6 \: is \:a \: rational \: number$