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Mathematics

Fill in the blanks to make the given expression a perfect square:

(i) 16a2 + 9b2 + …………..

(ii) 25a2 + 16b2 - …………..

(iii) 4a2 + 20ab + …………..

(iv) 9a2 - 24ab + …………..

Expansions

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Answer

(i) Given,

16a2 + 9b2 + …………..

Adding 24ab to above equation, we get :

⇒ 16a2 + 9b2 + 24ab

⇒ (4a)2 + (3b)2 + 2 × 4a × 3b

We know that,

(a + b)2 = a2 + b2 + 2ab

⇒ (4a + 3b)2.

Hence, on adding 24ab to the expression 16a2 + 9b2, it becomes a perfect square.

(ii) Given,

25a2 + 16b2 + …………..

Adding 40ab to above equation, we get :

⇒ 25a2 + 16b2 + 40ab

⇒ (5a)2 + (4b)2 + 2 × 5a × 4b

We know that,

(a + b)2 = a2 + b2 + 2ab

⇒ (5a + 4b)2.

Hence, on adding 40ab to the expression 25a2 + 16b2, it becomes a perfect square.

(iii) Given,

4a2 + 20ab + …………..

Adding 25b2 to above equation, we get :

⇒ 4a2 + 25b2 + 20ab

⇒ (2a)2 + (5b)2 + 2 × 2a × 5b

We know that,

(a + b)2 = a2 + b2 + 2ab

⇒ (2a + 5b)2

Hence, on adding 25b2 to the expression 4a2 + 20ab, it becomes a perfect square.

(iv) Given,

9a2 - 24ab + …………..

Adding 16b2 to above equation, we get :

⇒ 9a2 + 16b2 - 24ab

⇒ (3a)2 + (4b)2 - 2 × 3a × 4b

We know that,

(a - b)2 = a2 + b2 - 2ab

⇒ (3a - 4b)2

Hence, on adding 16b2 to the expression 9a2 - 24ab, it becomes a perfect square.

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