Question

If a = 2 + √3, find the value of
a -1/a​

Answer 1

Answer:

2√3

Step-by-step explanation:

= > a = 2 + √3

= > 1/a = 1/( 2 + √3 )

On RHS divide as well as multiply by 2 - √3:

= > 1/a = ( 2 - √3 ) / ( 2 + √3 )( 2 - √3 )

= > 1/a = ( 2 - √3 ) / ( 2^2 - √3^2 )

= > 1/a = ( 2 - √3 ) / ( 4 - 3 )

= > 1/a = ( 2 - √3 ) / 1

= > 1/a = 2 - √3

Hence,

= > a - 1/a

= > 2 + √3 - ( 2 - √3 )

= > 2 + √3 - 2 + √3

= > √3 + √3

= > 2√3

Answer 2

Answer:

→ a = 2 + √3

$\dfrac{1}{a} =\dfrac{1}{2+\sqrt3}$

Multiply and divide RHS by 2 - √3

$\implies\dfrac{1}{a} =\dfrac{1}{2+\sqrt3}\times\dfrac{2-\sqrt3}{2-\sqrt3}\\\\\\\implies\dfrac{1}{a} =\dfrac{2-\sqrt3}{(2+\sqrt3)(2-\sqrt3)}$

Using ( a + b )( a - b ) = a² - b²

$\implies\dfrac{1}{a} =\dfrac{2-\sqrt3}{2^2 - (\sqrt3)^2}\\\\\\\implies\dfrac{1}{a} =\dfrac{2-\sqrt3}{4-3}\\\\\\\implies \dfrac{1}{a} =2-\sqrt3$

Hence,

a - 1/a = 2 + √3 - ( 2 - √3 ) = 2 + √3 - 2 + √3 = 2√3