Question

if X=(√3-√2) then find the value of (x-1/X)^2​

Answer 1

Answer:

Step-by-step explanation:

Given that,

x = ( √3 - √2 )

Now squaring on both sides, we get

x^2 = ( √3 - √2 )^2

= ( √3 )^2 + ( √2 )^2 - 2 × √3 × √2 ( ∵ (a -b )^2 = a ^2 + b ^2 - 2ab )

= 3 + 2 - 2√6 ( ∵ roots and square 2's are cancelled )

= 5 - 2√6

∴ x^2 = 5 - 2√6

Now we find (1 / x ^2)

1 / x ^2 = 1 / (5 - 2√6 )

= Rationalize the denominator, we get

= 1 / ( 5 - 2√6 ) × (5 + 2√6 ) / ( 5 + 2√6 )

= 1 × ( 5 +2√6 ) / (5 - 2√6) × (5 + 2√6)

= (5 + 2√6) / ( 5 )^2 - (2√6)^2 ( ∵ (a-b)(a+b) = a^2 - b^2 )

=(5 + 2√6 ) / 25 - (2)^2 × (√6)^2

= (5+ 2√6) / 25 - 4 × 6 ( ∵root and square 2 are cancelled )

= ( 5 + 2√6 ) / 25 - 24

= ( 5 + 2√6 ) / 1

= 5 + 2√6

∴ 1 / (x )^2 = 5 + 2√6

Now we find ( x -1 / x )^2

( x - 1 /× )^2 = (x )^2 + ( 1 / x )^2 - 2 × (x ) × (1 / x )

= ( 5 - 2√6 ) + ( 5 + 2√6 ) - 2 ( ∵ x and 1/x are cancelled )

= 5 - 2√6 + 5 + 2√6 - 2

= 5 + 5 - 2 ( ∵ -2√6 and +2√6 are cancelled )

= 10 - 2

= 8

∴ ( x - 1 / x )^2 = 8 is the answer.