Question

If = 2−√5 2+√5 and = 2+√5 2−√5 , find a^2 + b^2

Answer 1

I think question is like this

a = ( 2 - √5 )/ (2 + √5 ) and

b = ( 2 + √5 ) / (2 - √5 ) then find a² + b²

Solution --->

a = (2 - √5 ) / ( 2 + √ 5 )

Now multiplying in numerator and denominator by ( 2 - √5 ) we get

a = (2 - √5 )(2 - √5 )/(2 + √5 ) ( 2 - √5 )

= ( 2 - √ 5 )² / (2)² - (√5)²

We have a identity a² - b² =(a + b)(a -b )

applying it here we get.

= ( 2 - √5 )² / ( 4 - 5 )

We have an identity

(a - b )² = a² + b² -2ab ,applying it

= (2)² + (√5 )² - 2 (2)(√5) / (-1)

= - (4 + 5 - 4√5)

= - (9 - 4√5)

= 4√5 - 9

Now

b = ( 2+√5) / (2- √5)

Multiplying by (2 + √5 ) in numerator and denominator we get

b= (2 + √5)(2 + √5) / (2 - √5)(2 + √5)

= ( 2 + √5 )² / (2)² - (√5)²

We have an identity

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

= (2)² +(√5)² +2(2)(√5) /( 4 - 5)

= 4 + 5 + 4 √5 / (-1)

= - ( 9 + 4 √5 )

Now

a² + b² = (4√5 - 9)² + { - (9 + 4 √5) }²

=(4√5)² + (9)² - 2(4√5)(9) +(9)² +(4√5)²

+2(9) (4√5)

=80 + 81 -72√5 +81 + 80 + 72√5

-72√5 and +72√5 cancel out each other

and we get

= 322

Answer 2

Answer:

Step-by-step explanation:

if

a = (2-√5)/(2+√5)

b = (2+√5)/(2-√5)

⇒ a = (2-√5)/(2+√5) × (2-√5)/(2-√5)

⇒ (4-4√5+5)/(4-5)

⇒ (9-4√5)/-1

⇒ a = -9+4√5

⇒ b = (2+√5)/(2-√5) × (2+√5)/(2+√5)

⇒ (4+4√5+5)/(4-5)

⇒ (9+4√5)/-1

⇒ b = -9-4√5

⇒ a² - b² = (-9+4√5)² - (-9-4√5)

⇒ 81+80-72√5 - (81+80+72√5)

⇒ 81+80-72√5-81-80-72√5

⇒ -72√5-72√5

               a² - b² = -144√5