Question

If x=2+√3 find the value of x^2+1/x^2

Answer 1

The trick is to use the identity$x^2+\frac{1}{x^2}=\left(x+\frac{1}{x}\right)^2-2$

In view of using above identity, start by finding the value of $x+\frac{1}{x}$ :
$x=2+\sqrt{3}\\~\\\frac{1}{x}=\frac{1}{2+\sqrt{3}}=\frac{1}{2+\sqrt{3}}\times\frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{2^2-\sqrt{3}^2}=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3}$

Adding these
$x+\frac{1}{x}=2+\sqrt{3}+2-\sqrt{3}=4$

Plug in the earlier identity and get
$x^2+\frac{1}{x^2}=4^2-2=14$