Question

If a=3-2√2, find a ²-1/a^2​

Answer 1

Answer:

34

Step-by-step explanation:

Given-----> a = 3 - 2√2

To find----> a² - 1 / a²

Solution-----> ATQ,

a = 3 - 2√2

1 / a = 1 / ( 3 - 2√2 )

Multiplying by ( 3 + 2√2 ) in numerator and denominator we get,

= ( 3 + 2√2 ) / ( 3 - 2√2 ) ( 3 + 2√2 )

We know that, a² - b² = ( a + b ) ( a - b ) , we get,

= ( 3 + 2√2 ) / { ( 3 )² - ( 2√2 )² }

= ( 3 + 2√2 ) / ( 9 - 8 )

= ( 3 + 2√2 ) / 1

=> 1/a = ( 3 + 2√2 )

Now , a = ( 3 - 2√2 )

Squaring both sides , we get,

=> a² = ( 3 - 2√2 )²

We know that,

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

= ( 3 )² + ( 2√2 )² - 2 ( 3 ) ( 2√2 )

= 9 + 8 - 12√2

=> a² = 17 - 12√2

Now, 1 / a = 3 - 2√2

Sqaring both sides we get,

=> 1 / a² = ( 3 + 2√2 )²

We know that ,

( a + b )² = a² + b² + 2ab , we get,

= ( 3 )² + ( 2√2 )² + 2 ( 3 ) ( 2√2 )

= 9 + 8 + 12√2

=> 1 / a² = 17 + 12√2

Now,

a² + 1 / a² = ( 17 - 12√2 ) + ( 17 + 12√2 )

= 34