Question

$if \: x + \frac{1}{x} = \sqrt{5} \: \: find \: the \: value \: of \: {x}^{2} + \frac{1}{ {x}^{2} } \: and \: {x}^{4} + \frac{1}{ {x}^{4} }$
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Answer 1

The answer for the given riddle is pencil.

Explanation:

The hints for the riddle are iam from a mine and wood is major hint.

Generally natural resources like coal, diamond, graphite, iron are from mines.

In the above natural resources generally graphite is surrounded by wood and daily used by us as pencil for writing.

Answer 2

Step-by-step explanation:

$x + \frac{1}{x} = \sqrt{5} \\ \\ = > {(x + \frac{1}{x} )}^{2} = ({ \sqrt{5} })^{2} \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2.x. \frac{1}{x } = 5 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 5 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = 5 - 2 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = 3 \\ \\ = > ({x}^{2} + \frac{1}{ {x}^{2} } ) {}^{2} = ({3})^{2} \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } + 2. {x}^{2} . \frac{1}{ {x}^{2} } = 9 \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 9 \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = 9 - 2 \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = 7 \\ \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 3 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 7 \\ \\ \\ \\ using \: formula \: (x + y) {}^{2} = {x}^{2} + {y}^{2} + 2 \times x \times y \\ \\ {( \sqrt{x} )}^{2} = x \: \: \: \:$