Question

if x=(3+√8) find the value of (x^2+1÷x^2)
​

Answer 1

Given:-

$\tt \rightarrow x=3+\sqrt{8}$

To find:-

$\tt \rightarrow x^2+\dfrac{1}{x^2}$

Solution:-

First we have to find 1/x

So, finding 1/x

$\tt \dfrac{1}{3+\sqrt{8}} \\\\ \tt Rationalizing\: \:the\:\: denominator \\\\ \tt \dfrac{1}{3+\sqrt{8}}\times \dfrac{3-\sqrt{8}}{3-\sqrt{8}} \\\\ \tt \Rightarrow \dfrac{3-\sqrt{8}}{(3+\sqrt{8})(3-\sqrt{8})} \\\\ \tt \Rightarrow \dfrac{3-\sqrt{8}}{(3)^2-(\sqrt{8})^2} \\\\ \tt \Rightarrow \dfrac{3-\sqrt8}{9-8}\\\\ \tt \Rightarrow 3-\sqrt8$

So, we get 1/x = 3-√8

So, finding

$\tt \rightarrow x^2+\dfrac{1}{x^2}$

$\tt (3+\sqrt8)^2+(3-\sqrt8)^2\\\\ \tt = (3)^2+2(3)(\sqrt8)+(\sqrt{8})^2+(3)^2-2(3)(\sqrt8)+(\sqrt{8})^2 \\\\ \tt = (3)^2+\cancel{2(3)(\sqrt8)}+(\sqrt{8})^2+(3)^2\cancel{-2(3)(\sqrt8)}+(\sqrt{8})^2 \\\\ \tt =2(3)^2+2(\sqrt8)^2 \\\\ \tt = 18+16 \\\\ \tt = 34$

So, we get the answer as 34

Answer 2

Answer:

(x^2+1÷x^2) = 34

Step-by-step explanation:

x = 3 + √8

1/x = 1/(3 + √8)

Rationalize the denominator

1/(3 + √8) × ( 3 - √8 )/( 3 - √8 )

( 3 - √8 ) / { (3)^2 - (√8)^2 }

( 3 - √8 ) / ( 9 - 8 )

( 3 - √8 ) / 1

3 - √8

1/x = 3 - √8

Now,

x^2 + 1/x^2

x^2 + (1/x)^2

(3 + √8)^2 + { (3 - √8)^2}

{ (3)^2 + (√8)^2 + 2×3×√8 + { (3)^2 + (√8)^2 - 2×3×√8 }

(3)^2 + (√8)^2 + (3)^2 + (√8)^2

9 + 8 + 9 + 8

34 Ans