If p = 2 - 5 2 + 5 and q = 2 + 5 2 - 5 , find the values of: p - q

Given, p = 2 - 5 2 + 5 and q = 2 + 5 2 - 5 , We have to find the value of p - q . p-q = 2 − 5 2 + 5 + 2 + 5 2 − 5 = ( 2 − 5 ) 2 ( 2 + 5 ) 2 ( 2 ) 2 − ( 5 ) 2 = ( 4 + 5 − 4 5 ) − ( 4 + 5 + 4 5 ) 4 − 5 = − 8 5 − 1 Therefore, p - q = 8 5

If p = 2 - 5 2 + 5 and q = 2 + 5 2 - 5 , find the values of: p - q

Given, p = 2 - 5 2 + 5 and q = 2 + 5 2 - 5 , We have to find the value of p - q . p-q = 2 − 5 2 + 5 + 2 + 5 2 − 5 = ( 2 − 5 ) 2 ( 2 + 5 ) 2 ( 2 ) 2 − ( 5 ) 2 = ( 4 + 5 − 4 5 ) − ( 4 + 5 + 4 5 ) 4 − 5 = − 8 5 − 1 Therefore, p - q = 8 5

If p = 2 - 5 2 + 5 and q = 2 + 5 2 - 5 , find the value of: p 2 + q 2

Given, p = 2 - 5 2 + 5 and q = 2 + 5 2 - 5 , We have to find the value of: p 2 + q 2 We know that p + q = 2 − 5 2 + 5 + 2 + 5 2 − 5 = ( 2 - 5 ) 2 + ( 2 + 5 ) 2 ( 2 ) 2 - ( 5 ) 2 = ( 4 + 5 − 4 5 ) + ( 4 + 5 + 4 5 ) 4 − 5 = 18 - 1 Therefore, p + q = - 18 ( p + q ) 2 = p 2 + q 2 + 2 p q Therefore, p q = 2 − 5 2 + 5 × 2 + 5 2 − 5 = 1 So, p + q = - 18 p 2 + q 2 = ( p + q ) 2 - 2 p q = ( - 18 ) 2 - 2 × 1 = 324 - 2 = 322 Hence, p 2 + q 2 = 322

If p = 2 - 5 2 + 5 and q = 2 + 5 2 - 5 , find the value of: p 2 - q 2

Given, p = 2 - 5 2 + 5 and q = 2 + 5 2 - 5 , we have to find the value of: p 2 - q 2 We know that p - q = 2 − 5 2 + 5 - 2 + 5 2 − 5 = ( 2 - 5 ) 2 + ( 2 + 5 ) 2 ( 2 ) 2 - ( 5 ) 2 = ( 4 + 5 − 4 5 ) + ( 4 + 5 + 4 5 ) 4 − 5 = 18 - 1 Therefore, p + q = - 18 p - q = 2 − 5 2 + 5 + 2 + 5 2 − 5 = ( 2 − 5 ) 2 ( 2 + 5 ) 2 ( 2 ) 2 − ( 5 ) 2 = ( 4 + 5 − 4 5 ) − ( 4 + 5 + 4 5 ) 4 − 5 = − 8 5 − 1 Therefore, p - q = 8 5 p 2 - q 2 = ( p + q ) ( p - q ) = ( - 18 ) ( 8 5 ) = 144 5 Hence, p 2 - q 2 = 144 5

if x = 2 - 1 2 + 1 and y = 2 + 1 2 - 1 then find the value of x 2 + 5 x y + y 2

Given, x = 2 - 1 2 + 1 and y = 2 + 1 2 - 1 We have to find the value of x 2 + 5 x y + y 2 x 2 + 5 x y + y 2 = ( x + y ) 2 + 3 x y ⇒ 2 - 1 2 + 1 + 2 + 1 2 - 1 2 + 3 2 - 1 2 + 1 2 + 1 2 - 1 ⇒ 2 + 1 - 2 2 + 2 + 1 + 2 2 2 - 1 2 + 3 2 - 1 2 - 1 ⇒ 36 + 3 = 39 Hence, x 2 + 5 x y + y 2 = 39