Question

if a=1-√5 find a^2-1\a^2​

Answer 1

Step-by-step explanation:

$a = 1 - \sqrt{5} \\ \frac{1}{a} = \dfrac{1}{1 - \sqrt{5} } \\ = \dfrac{1}{1 - \sqrt{5} } \\ = \dfrac{1(1 + \sqrt{5}) }{(1 - \sqrt{5})(1 + \sqrt{5}) } \\ = \dfrac{1 + \sqrt{5} }{ {(1)}^{2} - {( \sqrt{5}) }^{2} } \\ = \dfrac{1 + \sqrt{5} }{1 - 5 } \\ = \dfrac{1 + \sqrt{5} }{ - 4}$

${a}^{2} - {( \dfrac{1}{a}) }^{2} \\ = {(1 - \sqrt{5} )}^{2} - {( \frac{1 + \sqrt{5} }{ - 4} )}^{2} \\ = {(1)}^{2} + {( \sqrt{5}) }^{2} - 2 \sqrt{5} - {( \frac{ {(1)}^{2} + { (\sqrt{5} )}^{2} + 2 \sqrt{5} }{16} )} \\ = 1 + 5 - 2 \sqrt{5} - \dfrac{1 + 5 + 2 \sqrt{5} }{16} \\ = 6 - 2 \sqrt{5} - \frac{6 + 2 \sqrt{5} }{16} \\ = \dfrac{96 - 32 \sqrt{5} - 6 - 2 \sqrt{5} }{16} \\ = \dfrac{90 - 32 \sqrt{5} }{16} \\ = \dfrac{2(45 - 16 \sqrt{5} )}{16} \\ = \dfrac{45 - 16 \sqrt{5} }{16}$

Answer 2

Answer:

a² - ( 1 / a² ) = ( 45 - 17 √5 ) / 8

Step-by-step-explanation:

We have given that,

a = 1 - √5

We have to find the value of a² - ( 1 / a² ).

Now,

1 / a = 1 / ( 1 - √5 )

By multiplying and dividing RHS by ( 1 + √5 ),

1 / a = [ 1 / ( 1 - √5 ) ] * ( 1 + √5 ) / ( 1 + √5 )

⇒ 1 / a = ( 1 + √5 ) / ( 1 - √5 ) ( 1 + √5 )

We know that,

( a + b ) ( a - b ) = a² - b²

⇒ 1 / a = ( 1 + √5 ) / [ ( 1 )² - ( √5 )² ]

⇒ 1 / a = ( 1 + √5 ) / ( 1 - 5 )

⇒ 1 / a = ( 1 + √5 ) / ( - 4 )

⇒ 1 / a = - ( 1 + √5 ) / 4

⇒ 1 / a = ( - 1 - √5 ) / 4

Now, we know that,

a² - b² = ( a + b ) ( a - b )

∴ a² - ( 1 / a² ) = [ a + ( 1 / a ) ] [ a - ( 1 / a ) ]

⇒ a² - ( 1 / a² ) = [ 1 - √5 + ( - 1 - √5 ) / 4 ] [ 1 - √5 - ( - 1 - √5 ) / 4 ]

⇒ a² - ( 1 / a² ) = { [ 4 ( 1 - √5 ) + ( - 1 - √5 ) ] / 4 } { [ 4 ( 1 - √5 ) - ( - 1 - √5 ) ] / 4 }

⇒ a² - ( 1 / a² ) = [ ( 4 - 4 √5 - 1 - √5 ) / 4 ] [ 4 - 4 √5 + 1 + √5 ) / 4 ]

⇒ a² - ( 1 / a² ) = [ ( 3 - 5 √5 ) / 4 ] [ ( 5 - 3 √5 ) / 4 ]

⇒ a² - ( 1 / a² ) = ( 3 - 5 √5 ) ( 5 - 3 √5 ) / ( 4 * 4 )

⇒ a² - ( 1 / a² ) = [ 3 ( 5 - 3 √5 ) - 5 √5 ( 5 - 3 √5 ) / 16

⇒ a² - ( 1 / a² ) = ( 15 - 9 √5 - 25 √5 + 15 * 5 ) / 16

⇒ a² - ( 1 / a² ) = ( 15 - 9 √5 - 25 √5 + 75 ) / 16

⇒ a² - ( 1 / a² ) = ( 15 + 75 - 9 √5 - 25 √5 ) / 16

⇒ a² - ( 1 / a² ) = ( 90 - 34 √5 ) / 16

⇒ a² - ( 1 / a² ) = 2 ( 45 - 17 √5 ) / 16

⇒ a² - ( 1 / a² ) = ( 45 - 17 √5 ) / 8