Question

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

Answer 1

• Given:

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

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

$\sf \implies \frac{3 - 2 \sqrt{2} }{9 - 8} = 3 - 2 \sqrt{2}$

$\sf \therefore \: x + \frac{1}{x} = 3 + 2 \sqrt{2} + 3 - 2 \sqrt{2} \\ \\ \sf = 6 \: which \: is \: a \: rational \: number$

✭Hope it helps you...

Answer 2

answer:

given ,x = 3+2✓2

x+1/x = ( 3+2✓2) + 1/(3+2✓2)

= (3+2✓2) +1/ 3+2✓2 × 3-2✓2/3-2✓2

[ by rationalisation]

= 3+2✓2 + 3-2✓2/9-8

[ (a+b) (a-b) = a^2 - b^2]

= 3+2✓2 + 3-2✓2/1 = 6

hence, x+1/x is rational