Question

If x=5-2√6 then find

1. 1/x
2. x-1/x
3. x+1/x

Answer 1

`\large\boxed{1.\ (1)/(x)=5+2\sqrt6}\\\boxed{2.\ (x-1)/(x)=-4-2\sqrt6}\\\boxed{3.\ (x+1)/(x)=6+2\sqrt6}`

Explanation:

`x=5-2\sqrt6\\\\1.\\\\(1)/(x)=(1)/(5-2\sqrt6)=(1)/(5-2\sqrt6)\cdot(5+2\sqrt6)/(5+2\sqrt6)\qquad\text{use}\ (a-b)(a+b)=a^2-b^2\\\\=(5+2\sqrt6)/(5^2-(2\sqrt6)^2)=(5+2\sqrt6)/(25-2^2(\sqrt6)^2)=(5+2\sqrt6)/(25-(4)(6))=(5+2\sqrt6)/(25-24)\\\\=(5+2\sqrt6)/(1)=5+2\sqrt6\\\\2.\\\\(x-1)/(x)=(x)/(x)-(1)/(x)=1-(1)/(x)\\\\\text{use the value of}\ (1)/(x)\ \text{from 1.}\\\\(x-1)/(x)=1-(5+2\sqrt6)=1-5-2\sqrt6=-4-2\sqrt6`

`3.\\\\(x+1)/(x)=(x)/(x)+(1)/(x)=1+(1)/(x)\\\\\text{use the value of}\ (1)/(x)\ \text{from 1.}\\\\(x+1)/(x)=1+5+2\sqrt6=6+2\sqrt6`