Question

if x=2-√5find x^2+1/x^2

Answer 1

Hey there!

Given,

$x = 2 - \sqrt{5}$

Find

$x {}^{2} + \frac{1}{x {}^{2} }$

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

using expression (a+b)² = a² + 2ab + b²

$4 - 4 \sqrt{5} + 5 + \frac{1}{4 - 4 \sqrt{5} + 5 }$

$4 - 4 \sqrt{5} + 5 + \frac{1}{9 - 4 \sqrt{5} }$

Rationalising the denominator

$\frac{1(9 + 4 \sqrt{5}) }{(9 - 4 \sqrt{5}) (9 + 4 \sqrt{5}) }$

$\frac{9 + 4 \sqrt{5} }{81 - 80}$

$9 + 4 \sqrt{5}$

$4 - 4 \sqrt{5} + 5 + 9 + 4 \sqrt{5}$

(cancelling the opposites)

$4 + 5 + 9$

$18$

Hope it helps!