Question

if x=3+√8,show that (x^2+1/x^2)=34

Answer 1

$\mathrm{x=3+\sqrt{8}} \\\\ \mathrm{x^2+ \frac{1}{x^2} =..} \\\\ \mathrm{\frac{1}{x^2} = \frac{1}{(3+\sqrt8)^2}} \\\ ~~~~~\mathrm{= \frac{1}{9+6\sqrt8+8} } \\\ ~~~~~\mathrm{= \frac{1}{17+6.2\sqrt2} } \\\ ~~~~=\mathrm{ \frac{1}{17+12\sqrt{2}} \Rightarrow multiply~root} \\\ ~~~~~\mathrm{= \frac{1}{17+12\sqrt{2}} ~x~ \frac{17-12\sqrt{2}}{17-12\sqrt{2}} } \\\ ~~~~~\mathrm{= \frac{17-12\sqrt{2}}{289-288} } \\\ ~~~~=\mathrm{17-12\sqrt{2}}$

$x^2 =(3+\sqrt8)^2 \\\ ~~~~=9+6\sqrt8+8 \\\ ~~~~=17+6.2\sqrt2 \\\ ~~~~=17+12\sqrt2 \\\ \mathrm{So,x^2+ \frac{1}{x^2} ~is.. } \\\ \mathrm{x^2+ \frac{1}{x^2} =(17+12\sqrt2)+(17-12\sqrt2)} \\\ \mathrm{ x^2+ \frac{1}{x^2} =17+17} \\\ \mathrm{x^2+ \frac{1}{x^2} =34} \\\ \bigstar~\mathrm{SOLVED}~\bigstar$