Write down the square of the expression 5 p - 1 4 q .

Given: 5 p - 1 4 q The square of 5 p - 1 4 q is given by 5 p - 1 4 q 2 . We know that, a - b 2 = a 2 - 2 a b + b 2 Here, a = 5 p and b = 1 4 q So, substituting, ⇒ 5 p - 1 4 q 2 = 5 p 2 - 2 5 p 1 4 q + 1 4 q 2 ⇒ 5 p - 1 4 q 2 = 25 p 2 - 5 p 2 q + 1 16 q 2 Hence, 5 p - 1 4 q 2 = 25 p 2 - 5 p 2 q + 1 16 q 2

Write down the square of the expression 2 x - 1 3 p + 3 q .

Given: 2 x - 1 3 p + 3 q The square of 2 x - 1 3 p + 3 q is given by 2 x - 1 3 p + 3 q 2 . We know that, a + b + c 2 = a 2 + b 2 + c 2 + 2 a b + 2 b c + 2 c a Here, a = 2 x , b = - 1 3 p and c = 3 q So, substituting, ⇒ 2 x - 1 3 p + 3 q 2 = 2 x 2 + - 1 3 p 2 + 3 q 2 + 2 2 x - 1 3 p + 2 - 1 3 p 3 q + 2 3 q 2 x ⇒ 2 x - 1 3 p + 3 q 2 = 4 x 2 + 1 9 p 2 + 9 q 2 - 4 3 x p - 2 p q + 12 x q Hence, 2 x - 1 3 p + 3 q 2 = 4 x 2 + 1 9 p 2 + 9 q 2 - 4 3 x p - 2 p q + 12 x q

Expand 1 3 - 5 x 1 3 + 5 x .

We need to expand 1 3 - 5 x 1 3 + 5 x . We know the identity, ( a + b ) ( a - b ) = a 2 - b 2 Here, a = 1 3 and b = 5 x Substituting the values, ⇒ 1 3 - 5 x 1 3 + 5 x = 1 3 2 - 5 x 2 ⇒ 1 3 - 5 x 1 3 + 5 x = 1 9 - 25 x 2 Hence, 1 3 - 5 x 1 3 + 5 x = 1 9 - 25 x 2