Question

if x=7+√5 find the value of x+1/x

full explanation needed​

Answer 1

Answer:

$\large \text{ (x+\dfrac{1}{x})= \dfrac{301+43\sqrt{5}}{44} }$

Step-by-step explanation:

Given :

$\large x=7+\sqrt5$

Now taking reciprocal of both side

$\large \dfrac{1}{x}=\dfrac{1}{7+\sqrt5}$

Now rationalize the denominator

$\large \dfrac{1}{x}=\dfrac{1}{7+\sqrt5}\times\dfrac{7-\sqrt5}{7-\sqrt5} \\\\\\\large \dfrac{1}{x}=\dfrac{7-\sqrt5}{49-5}\\\\\\\large \dfrac{1}{x}=\dfrac{7-\sqrt5}{44}\\\\\\\large \text{Here we used identity a^2-b^2=(a+b)(a-b) }$

We have to find  $\large (x+\dfrac{1}{x})$

Now adding both

$\large x+\dfrac{1}{x}= 7+\sqrt5+\dfrac{7-\sqrt5}{44}\\\\\\\large x+\dfrac{1}{x}= \dfrac{7\times44+\sqrt5\times44+7-\sqrt5}{44}\\\\\\\large \text{ x+\dfrac{1}{x}= \dfrac{308-7+44\sqrt{44}-\sqrt5}{44} }\\\\\\\large \text{ x+\dfrac{1}{x}= \dfrac{301+43\sqrt{5}}{44} }$

Thus we get answer.

Answer 2

Step-by-step explanation:

x=7+√5

1/x=1/7+√5

1/x= 7-√5

(7+√5)(7-√5)

1/x=7-√5

44

now put value of 1/x then and

putting value of x=7+√5 then

7+√5+7-√5

44

308+44√5+7-√5

44

301+43√5

44