Question

find x2+1/x2,if x=1/2+✓5​

Answer 1

Answer:

23

Step-by-step explanation:

x = 1/2+√5

multiply numerator and denominator by 2-√5

x = 2 - √5/ (2 +√5)(2 - √5) = 2 - √5/ 4 - 5 = -2+ √5

1/x = 2+√5

x² + 1/x² = (x + 1/x)² -  2

              = (-2+√5 + 2 + √5)² - 2

               = 25 - 2 = 23.

Answer 2

$x=\dfrac{1}{2+\sqrt5}=\dfrac{\sqrt5-2}{(2+\sqrt5)(\sqrt5-2)}=\dfrac{\sqrt5-2}{5-4}=\sqrt5-2$

$\dfrac{1}{x}=2+\sqrt5$

So,

$\begin{aligned}&x-\dfrac{1}{x}=(2+\sqrt5)-(\sqrt5-2)\\\\\implies\ \ &\left(x-\dfrac{1}{x}\right)^2=4^2\\\\\implies\ \ &x^2+\dfrac{1}{x^2}-2=16\\\\\implies\ \ &x^2+\dfrac{1}{x^2}=\mathbf{18}\end{aligned}$