Question

If x+1/x=√5, then find x^2+1/x^2 & x^4+1/x^4

Answer 1

Heya friend,
Here is the answer you were looking for:

Identity used :

${(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab$

$x + \frac{1}{x} = \sqrt{5}$

On squaring both the side we get,

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

Again on squaring both the sides we get,


${( {x}^{2} + \frac{1}{ {x}^{2} } )}^{2} = {(3)}^{2} \\ \\ {( {x}^{2} )}^{2} + {( \frac{1}{ {x}^{2} } )}^{2} + 2 \times {x}^{2} \times \frac{1}{ {x}^{2} } = 9 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 9 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 9 - 2 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 7$

Hope this helps!!!

Feel free to ask in the comment section if you have any doubt regarding to my answer...

@Mahak24

Thanks...
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