Question

If x=7+5√2 then find the value of x^2+1/x^2

Answer 1

GIVEN :

x = 7 + 5√2

TO FIND :

Value of x² + 1/x²

SOLUTION :

x = 7 + 5√2

1/x = 1/(7 + 5√2)

Rationalizing

→ 1/x = 1/(7 + 5√2) × (7 - 5√2)/(7 - 5√2)

→ 1/x = (7 - 5√2)/(49 - 50)

→ 1/x = (7 - 5√2)/- 1 = 5√2 - 7

(x + 1/x)² - 2 = x² + 1/x²

→ (7 + 5√2 + 5√2 - 7)² - 2

→ (10√2)² - 2

→ 200 - 2 = 198

x² + 1/x² = 198

Therefore, value of x² + 1/x² = 198.

Answer 2

$\bigstar$ 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

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

Now by rationalising the denominator,

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

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

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

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

( 5√2 × 5√2 = 5 × 5 × 2 = 50 )

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

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

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

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

$\implies$ $\boxed{x ^{2} \:+ \dfrac{1}{x ^{2} }\:= 198}$

Answer:

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