Question

If $x=9-4\sqrt{5}$ then, find the value of $x+\frac{14}{x}$

Answer 1

$\begin{array}{c}wehave\\ x=9-4\sqrt{4}\\ so,\\ \frac{1}{x}=\frac{1}{9-4\sqrt{5}}\\ =\frac{1}{9-4\sqrt{5}}\times \frac{9+4\sqrt{5}}{9+4\sqrt{5}}\\ =\frac{9+4\sqrt{5}}{81-80}\\ =9+4\sqrt{5}\\ Now.\\ x+\frac{14}{x}=9-4\sqrt{5}+14\left(9+4\sqrt{5},\right)\\ =9-4\sqrt{5}+126+56\sqrt{5}\\ =135+52\sqrt{5}\\ Hence,therequiredansweris135+52\sqrt{5}..\end{array}$