If x = 5 - 3 5 + 3 and y = 5 + 3 5 - 3 show that x - y = - 10 3 11

Given, x = 5 - 3 5 + 3 So, x = 5 - 3 5 + 3 × 5 - 3 5 - 3 [According to the formula a 2 - b 2 = ( a + b ) ( a - b ) ] = ( 5 - 3 ) 2 5 2 - 3 2 = 25 - 10 3 + 3 25 - 3 = 28 - 10 3 22 = 14 - 5 3 11 . Also, given, y = 5 + 3 5 - 3 So, y = 5 + 3 5 - 3 × 5 + 3 5 + 3 [According to the formula a 2 - b 2 = ( a + b ) ( a - b ) ] = ( 5 + 3 ) 2 5 2 - 3 2 = 25 + 10 3 + 3 25 - 3 = 28 + 10 3 22 = 14 + 5 3 11 . Now, x - y = 14 - 5 3 11 - 14 + 5 3 11 = 14 - 5 3 - 14 - 5 3 11 = - 10 3 11 (proved).

If a = 5 + 2 5 - 2 and b = 5 - 2 5 + 2 show that 3 a 2 + 4 a b - 3 b 2 = 4 + 56 3 10

Given: a = 5 + 2 5 - 2 Since denominator is an irrational number we have to rationalize it by multiplying its numerator and denominator by its rationalizing factor we get So, a = 5 + 2 5 - 2 × 5 + 2 5 + 2 [According to the formula a 2 - b 2 = ( a + b ) ( a - b ) ] = ( 5 + 2 ) 2 5 2 - 2 2 = 5 + 2 10 + 2 5 - 2 = 7 + 2 10 3 . Also, given that, b = 5 - 2 5 + 2 Since denominator is an irrational number we have to rationalize it by multiplying its numerator and denominator by its rationalizing factor we get b = 5 - 2 5 + 2 × 5 - 2 5 - 2 [According to the formula a 2 - b 2 = ( a + b ) ( a - b ) ] = ( 5 - 2 ) 2 5 2 - 2 2 = 5 - 2 10 + 2 5 - 2 = 7 - 2 10 3 . a 2 = 7 + 2 10 3 2 = 49 + 28 10 + 40 9 = 89 + 28 10 9 . b 2 = 7 - 2 10 3 2 = 49 - 28 10 + 40 9 = 89 - 28 10 9 . Now, L H S = 3 a 2 + 4 a b - 3 b 2 = 3 × 89 + 28 10 9 + 4 × 5 + 2 5 - 2 × 5 - 2 5 + 2 - 3 × 89 - 28 10 9 = 89 + 28 10 3 + 4 - 89 - 28 10 3 = 4 + 89 + 28 10 - 89 + 28 10 3 = 4 + 56 3 10 = RHS . (proved)