Question

if x=6-sqrt(35) find(1)/(x),x+(1)/(x),x^2+(1)/(x^2)

Answer 1

Given,

x=6-sqrt(35)

1. (1)/(x)=1/(6-sqrt(35))

Rationalizing denominator,

(1)/(x)=

{1/(6-sqrt(35))}*{6+sqrt(35)/6+sqrt(35)}

ie, (1)/(x)=(6+sqrt(35)/6^2-sqrt(35)^2)

Since

$( a+ b)( a- b) = {a}^{2} - {b}^{2}$

We have,

(1)/(x)=6+sqrt(35)/(36-35)

(1)/(x)=6+sqrt(35)

2. x+(1)/(x)= 6-sqrt(35)+6+sqrt(35)

= 6+6

=12

3. x^2+(1)/(x^2)

We know that

${a}^{2} + {b}^{2} = {( a+ b)}^{2} - 2ab$

Here a=x, b=1/x

We have {x+(1)/(x)}^2 =12^2

=144

x^2+(1)/(x^2) = {x+(1)/(x)}^2 - (2*x*1/x)

=144-2

=142