Question

If x=(5+2√6),then show that x^2+1/x^2=?

Answer 1

$\huge{\pink{\underline{\ulcorner{\star\: Solution\: \star}\urcorner}}}$

$x =(5+2\sqrt{6}) \\\\ so, that \\\\ \frac{1}{x} = \frac{1}{5+2\sqrt{6}} ......[Eq \: 1]$

$\red{\underline{Rationalising\: equation\: first}}$

$\frac{1}{x} = \frac{(5-2\sqrt{6})}{(5+2\sqrt{6})(5-2\sqrt{6})}\\\\ \implies \frac{(5-2\sqrt{6})}{ (5)^{2} -(2\sqrt{6})^{2}} \\\\ \implies \frac{5-2\sqrt{6}}{25 - (4\times 6)} \\\\ \implies \frac{5-2\sqrt{6}}{25-24} \\\\ \implies \frac{5-2\sqrt{6}}{1} \\\\ \implies 5-2\sqrt{6}$

Now we have a values as follows

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

Putting the value in equation

$x^{2} + (\frac{1}{x})^{2} \\\\ (5+2\sqrt{6})^{2} + (5-2\sqrt{6})^{2} \\\\ 5^{2} + (2\sqrt{6})^{2} + \cancel{20\sqrt{6}} + 5^{2} + (2\sqrt{6})^{2} -\cancel{20\sqrt{6}} \\\\ 25 + 24 +25 +24 \\\\ 98$

$\red{\underline{Answer}}$

$\bold{\boxed{x^{2}+ \frac{1}{x^{2}} \: = \: 98}}$