If x + 1 x = 5 find the value of x 2 + 1 x 2 and x 4 + 1 x 4

In the given problem, we have to find x 2 + 1 x 2 and x 4 + 1 x 4 We have x + 1 x = 5 On squaring both sides we get, x + 1 x 2 = ( 5 ) 2 Using the algebraic identity: ( x + y ) 2 = x 2 + 2 x y + y 2 ⇒ x 2 + 1 x 2 + 2 × x × 1 x = 5 ⇒ x 2 + 1 x 2 = 3 Again squaring on both sides we get, x 2 + 1 x 2 2 = 3 2 x 4 + 1 x 4 + 2 × x 2 × 1 x 2 = 9 x 4 + 1 x 4 = 7 Hence, the value of x 2 + 1 x 2 is 3 and x 4 + 1 x 4 is 7

Simplify the following product. 1 2 a - 3 b 3 b + 1 2 a 1 4 a 2 + 9 b 2

Given: 1 2 a - 3 b 3 b + 1 2 a 1 4 a 2 + 9 b 2 On rearranging we get, 1 2 a - 3 b 1 2 a + 3 b 1 4 a 2 + 9 b 2 We shall use the identity ( x - y ) ( x + y ) = x 2 - y 2 1 2 a - 3 b 3 b + 1 2 a 1 4 a 2 + 9 b 2 = 1 2 a 2 - ( 3 b ) 2 1 4 a 2 + 9 b 2 = 1 4 a 2 - 9 b 2 1 4 a 2 + 9 b 2 Again using the algebraic identity: 1 2 a - 3 b 3 b + 1 2 a 1 4 a 2 + 9 b 2 = 1 4 a 2 - 9 b 2 1 4 a 2 + 9 b 2 = 1 4 a 2 2 - 9 b 2 2 = 1 16 a 4 - 81 b 4 Hence, the value of 1 2 a - 3 b 3 b + 1 2 a 1 4 a 2 + 9 b 2 is 1 16 a 4 - 81 b 4

Simplify the following product m + n 7 3 m - n 7

In the given problem, we have to find product of m + n 7 3 m - n 7 We have been given m + n 7 3 m - n 7 On rearranging we get m + n 7 3 m - n 7 = m + n 7 2 m + n 7 m - n 7 We shall use the identity ( x - y ) ( x + y ) = x 2 - y 2 m + n 7 3 m - n 7 = m + n 7 2 m 2 - n 7 2 = m + n 7 2 m 2 - n 2 49 Hence, the value of m + n 7 3 m - n 7 is m + n 7 2 m 2 - n 2 49

If x 2 + 1 x 2 = 66 find the value of x - 1 x

In the given problem, we have to find the value of x - 1 x Given x 2 + 1 x 2 = 66 Adding and subtracting 2 on left-hand side gives x 2 + 1 x 2 + 2 - 2 = 66 ⇒ x 2 + 1 x 2 - 2 × x × 1 x = 66 - 2 ⇒ x - 1 x 2 = 64 ⇒ x - 1 x 2 = 8 2 ⇒ x - 1 x = ± 8 Hence, the value of x - 1 x = ± 8