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

Question

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

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Amara 1 month 2021-08-14T14:05:40+00:00 1 Answer 0 views 0

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    2021-08-14T14:06:53+00:00

    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

    I hope it will help you

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