Question

If x + 2 =√5, find the value of x^4+1/x^4​

Answer 1

Answer:

$x + 2 = \sqrt{5} \\ x = \sqrt{5} - 2 \\ \: \: \: \: put \: \: the \: \: value \: \: in \: \\ \frac{ {x}^{4} + 1 }{ {x}^{4} } \\ = \frac{( \sqrt{5} - 2 {)}^{4} + 1 }{( \sqrt{5} - 2 {)}^{4} } \\ = \frac{(( \sqrt{5} - 2 {)}^{2} {)}^{2} + 1 }{(( \sqrt{5} - 2 {)}^{2} {)}^{2} } \\ now \: \: we \: \: will \: \: use \: \: this \: \: identity \\ (x - y {)}^{2} = {x}^{2} + {y}^{2} - 2xy \\ = \frac{(( \sqrt{5} {)}^{2} + {2}^{2} - 2 \times \sqrt{5} \times 2 {)}^{2} + 1 }{(( \sqrt{5 } {)}^{2} + {2}^{2} - 2 \times \sqrt{5} \times 2 {)}^{2} } \\ = \frac{(5 + 4 - 4 \sqrt{5}) + 1 }{(5 + 4 - 4 \sqrt{5}) } \\ = \frac{9 - 4 \sqrt{5} + 1 }{9 - 4 \sqrt{5} } \\ answer$

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