Question

If x+1/x=6 find x-1/x, x^2-1/x^2

Answer 1

Answer:

x - 1 / x = 4√2

x² - 1 / x² = 24√2

Step-by-step explanation:

Given-----> x + 1 / x = 6

To find-----> x - 1 / x , x² - 1 / x²

Solution------> ATQ,

x + 1 / x = 6

Squaring both sides , we get,

=> ( x + 1 / x )² = ( 6 )²

We have an identity,

( a + b )² = a² + b² + 2ab , applying it , we get,

=> x² + 1 / x² + 2 x ( 1 / x ) = 36

=> x² + 1 / x² + 2 = 36

=> x² + 1 / x² = 36 - 2

=> x² + 1 / x² = 34

Now,

( x - 1 / x )² = x² + 1 / x² - 2 ( x ) ( 1 / x )

= ( x² + 1 / x² ) - 2

= ( 34 ) - 2

=> ( x - 1 / x )² = 32

Taking square root of both sides , we get,

=> ( x - 1 / x ) = √32

= √( 16 × 2 )

=> ( x - 1 / x ) = 4 √2

Now,

x² - 1 / x² = ( x )² - ( 1 / x )²

We have an identity ,

( a² - b² ) = ( a + b ) ( a - b ) , applying it we get,

x² - 1 / x² = ( x + 1 / x ) ( x - 1 / x )

= ( 6 ) × ( 4√2 )

= 24√2