Question

If x= (3+√8), show that (x² + 1 / x² ) = 34​

Answer 1

Step-by-step explanation:

Given :-

$x = 3 + \sqrt{8}$

To Prove :-

${x}^{2} + \frac{1}{ {x}^{2} } = 34$

Proof :-

Here.

$x = 3 + \sqrt{8}$

then

$\frac{1}{x} = \frac{1}{3 + \sqrt{8} } \\ \\ = \frac{1}{3 + \sqrt{8} } \times \frac{3 - \sqrt{8} }{3 - \sqrt{8} } \\ \\ = \frac{3 - \sqrt{8} }{ {3}^{2} - ( { \sqrt{8}) }^{2} } \\ \\ = \frac{3 - \sqrt{8} }{9 - 8} = 3 - \sqrt{8} \\ \\ = > \frac{1}{x} = 3 - \sqrt{8} \: \: \: \: \: \: \: \: \: \:$

Now

$x + \frac{1}{x} = (3 + \sqrt{8} ) + (3 - \sqrt{8} ) \\ \\ x + \frac{1}{x} = 6$

Squaring both side

$(x + \frac{1}{x} ) {}^{2} = {6}^{2} \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2x \times \frac{1}{x} = 36 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 36 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = 36 - 2 = 34 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = 34$

Proved