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Mathematics

If a2 + b2 = 34 and ab = 12; find :

(i) 3(a + b)2 + 5(a - b)2

(ii) 7(a - b)2 - 2(a + b)2

Expansions

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Answer

(i) Expanding,

⇒ 3(a + b)2 + 5(a - b)2

⇒ 3(a2 + b2 + 2ab) + 5(a2 + b2 - 2ab)

⇒ 3(34 + 2 × 12) + 5(34 - 2 × 12)

⇒ 3(34 + 24) + 5(34 - 24)

⇒ 3 × 58 + 5 × 10

⇒ 174 + 50

⇒ 224.

Hence, 3(a + b)2 + 5(a - b)2 = 224.

(ii) Expanding,

⇒ 7(a - b)2 - 2(a + b)2

⇒ 7(a2 + b2 - 2ab) - 2(a2 + b2 + 2ab)

⇒ 7(34 - 2 × 12) - 2(34 + 2 × 12)

⇒ 7(34 - 24) - 2(34 + 24)

⇒ 7 × 10 - 2 × 58

⇒ 70 - 116

⇒ -46.

Hence, 7(a - b)2 - 2(a + b)2 = -46.

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