Question

ifx = 4+ √5 find the value of x² + 1/ x²​

Answer 1

hope it helps you.....to get your answer.....

Answer 2

62

Step-by-step explanation:

I can solve

1. Find the value of 1/x

2. Find the value of (x + 1/x)

3. Use Algebra Formula

4. Simplify till Answer

Given:

$x = 4 + \sqrt{15}$

To Find:

$\tt \: The \: value \: of \: {x}^{2} + \dfrac{1}{ {x}^{2} }$

Solution:

$x = 4 + \sqrt{15} \\ \\ \frac{1}{x} = \frac{1}{4 + \sqrt{15}} \\ \\ = \frac{(4 - \sqrt{15})}{(4 + \sqrt{15})(4 - \sqrt{15})}$

[Rationalizing The denominator]

$= \frac{4 - \sqrt{15}}{ {(4)}^{2} - {( \sqrt{15})}^{2} } \: \: \: [ \because \: {a}^{2} - {b}^{2} = (a + b)(a - b)] \\ \\ = \frac{4 - \sqrt{15} }{16 - 15} \\ \\ = \frac{4 - \sqrt{15} }{1} \\ \\ \therefore \: \frac{1}{x} = 4 - \sqrt{15} \\ \\ \tt \: So, \: x + \frac{1}{x} = (4 + \sqrt{15})+( 4 -\sqrt{15} ) \\ \\ \: x + \frac{1}{x} = 4 + 4 = 8$

Now,

${x}^{2} + \frac{1}{ {x}^{2} } \: \: \: [ \because \: {a}^{2} + {b}^{2} = {(a + b)}^{2} - 2ab]\\ \\ = (x + \frac{1}{x}) {}^{2} - 2.x. \frac{1}{x} \\ \\ = (x + \frac{1}{x} ) {}^{2} - 2 \\ \\ = {(8)}^{2} - 2 \\ \\ = 64 - 2 \\ \\ = 62$