Question

If $x=2\sqrt{6}+5,$ then find $x+\frac{1}{x}$

Answer 1

Given: $x=2\sqrt{6}+5$

$\frac{1}{x}=\frac{1}{2\sqrt{6}+5}\times \frac{2\sqrt{6}-5}{2\sqrt{6}-5}$

$=\frac{2\sqrt{6}-5}{{\left(2\sqrt{6}\right)}^2-5^2}$

$=\frac{2\sqrt{6}-5}{24-25}$

$=-2\sqrt{6}+5$

$\therefore x+\frac{1}{x}=2\sqrt{6}+5-2\sqrt{6}+5=10$

$\therefore x+\frac{1}{x}=10$