Question

If x=(7+5sqrt(2)) then (x^(2)+(1)/(x^(2)))=?​

Answer 1

$\bigstar$ Correct Question:

• If x = 7 + 5√2, then find the value of x² + 1/x².

$\bigstar$ Given:

• x = 7 + 5√2

$\bigstar$To find:

• x² + 1/x²

$\bigstar$ Solution:

x = 7 + 5√2

So, 1/x

$= \frac{1}{7 + 5 \sqrt{2} }$

Now by rationalising the denominator,

$\frac{1(7 - 5 \sqrt{2} )}{(7 + 5 \sqrt{2} )(7 - 5 \sqrt{2} )}$

$= \frac{7 - 5 \sqrt{2} }{ {7}^{2} - ( {5 \sqrt{2} )}^{2} }$

(By using ( a + b ) ( a - b ) = a² - b² )

$= \frac{7 - 5 \sqrt{2} }{49 - 50}$

( 5√2 × 5√2 = 5 × 5 × 2 = 50 )

$= \frac{7 - 5 \sqrt{2} }{( - 1)}$

$= - (7 - 5 \sqrt{2} )$

$= (- 7 )+ 5 \sqrt{2} \: \: or \: \: 5 \sqrt{2 } - 7$

So, x + 1/x = 5√2 - 7 + 7 + 5√2

$\implies$ x + 1/x = 10√2

Now,

x² + 1/x² can also be written as

( x + 1/x )² - 2.

x² + 1/x² = ( 10√2 )² - 2

$\implies$ x² + 1/x² = 200 - 2

$\implies$ $\boxed{x² + 1/x² = 198}$

$\bigstar$Answer:

$\therefore$ x² + 1/x² = $\boxed{198}$

Answer 2

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