Question

if x is equal to 2 + root 5 find the value of x square + 1 by x square please give this answer in step by step​

Answer 1

Answer:

$\frac{30}{29}$

Step-by-step explanation:

$x = 2 + \sqrt{5}$

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

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

$\frac{4 + 5 + 20 + 1}{4 + 5 + 20}$

$\frac{30}{29}$

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Answer 2

${x }^{2} + \frac{1}{ {x}^{2} } = 3$

Step-by-step explanation:

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

${(x + \frac{1}{x} )}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } + 2 \\ \\ {(2 + \sqrt{5} + \sqrt{5} - 2)}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } + 2 \\ \\ { (\sqrt{5} )}^{2} - 2 = {x}^{2} + \frac{1}{ {x}^{2} } \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 5 - 2 = 3$

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