Question

If x=5-2root5 find x^2+1/x^2

Answer 1

Answer:

answer 3 thousantand 14 thousant

Answer 2

Answer:

$x^{2}$ + $\frac{1}{x^{2}}$ = $\frac{234 - 96\sqrt{5} }{5}$

Step-by-step explanation:

$x = 5 - 2\sqrt{5}$

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

∴ $x$ + $\frac{1}{x}$ = $5 - 2\sqrt{5}$ + $\frac{5 + 2\sqrt{5} }{5}$ = $\frac{25 - 10\sqrt{5} + 5 + 2\sqrt{5} }{5}$ = $\frac{30 - 8\sqrt{5} }{5}$

$(x + \frac{1}{x} )^{2}$ = $\frac{900 - 2.30.8\sqrt{5} + 64.5 }{25}$ [squaring the whole equation]

=> $x^{2}$ + $2.x.\frac{1}{x}$ + $\frac{1}{x^{2}}$ =$\frac{900 + 320 - 480\sqrt{5} }{25}$

=> $x^{2}$ + $\frac{1}{x^{2}}$ = $\frac{900 + 320 - 480\sqrt{5} }{25} - 2$ = $\frac{1170 - 480\sqrt{5} }{25}$ = $\frac{234 - 96\sqrt{5} }{5}$