Question

If x(x-2)=1, then the x²-1/x² is ?​

Answer 1

Answer:

x² - 1 / x² = 4 √2

Step-by-step explanation:

Given---> x ( x - 2 ) = 1

To find ---> Value of

x² - 1 / x²

Solution---> x ( x - 2 ) = 1

Dividing equation by x we get

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

=> x - 2 = 1 / x

=> x - 1 / x = 2

Squaring both sides

=> ( x - 1 / x )² = ( 2 )²

We have an identity

( a - b )² = a² + b² - 2 ab

applying it here

=> x² + 1 / x² - 2 ( x ) ( 1 / x ) = 4

=> x² + 1 / x² - 2 = 4

=> x² + 1 / x² = 4 + 2

=> x² + 1 / x² = 6

Now

We have an identity

(a + b )² = a² + b² + 2 a b

Applying it here

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

= ( x² + 1 / x² ) + 2

= ( 6 ) + 2

( x + 1 / x )²= 8

Taking square root of both side

=> x + 1 / x = √8

=> x + 1 / x = 2√2

Now returning to original problem

x² - 1 / x² = ( x )² - ( 1/ x )²

We have an identity

a² - b² = ( a + b ) ( a - b )

Applying it here

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

Putting x + 1/x =2 √2 and x - 1/x = 2 in it

x² - 1 / x² = ( 2√2 ) ( 2 )

= 4 √2