Question

If x =2+√5 find x power 2 +1÷x power 2

Answer 1

Hey friend,
Here is the answer you were looking for:
$x = 2 + \sqrt{5} \\ \\ \frac{1}{x} = \frac{1}{2 + \sqrt{5} }$

On rationalizing the denominator we get,

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

Using the identity :

$(x +y)(x - y) = {x}^{2} - {y}^{2}$

$\frac{1}{x} = \frac{2 - \sqrt{5} }{ {(2)}^{2} - {( \sqrt{5} )}^{2} } \\ \\ \frac{1}{x} = \frac{2 - \sqrt{5} }{4 - 5} \\ \\ \frac{1}{x} = - 2 + \sqrt{5} \\ \\ x + \frac{1}{x} = (2 + \sqrt{5} ) + ( - 2 + \sqrt{5} ) \\ \\ x + \frac{1}{x} = 2 + \sqrt{5} - 2 + \sqrt{5} \\ \\ x + \frac{1}{x} = 2 \sqrt{5}$

Squaring both the sides :

${(x + \frac{1}{x}) }^{2} = {(2 \sqrt{5}) }^{2} \\ \\ {x}^{2} + {( \frac{1}{x} )}^{2} + 2 \times x \times \frac{1}{x} = 20 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 20 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 20 - 2 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 18$


Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
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