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Mathematics

Find the expansions of the following :

(i) (2x + 3y + 5)(2x + 3y - 5)

(ii) (6 - 4a - 7b)2

(iii) (7 - 3xy)3

(iv) (x + y + 2)3

Expansions

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Answer

(i) On solving,

⇒ (2x + 3y + 5)(2x + 3y - 5) = (2x + 3y)2 - 52

⇒ (2x + 3y + 5)(2x + 3y - 5) = 4x2 + 9y2 + 12xy - 25.

Hence, (2x + 3y + 5)(2x + 3y - 5) = 4x2 + 9y2 + 12xy - 25.

(ii) On solving,

⇒ (6 - 4a - 7b)2 = (6 - 4a)2 + (7b)2 - 2(6 - 4a)(7b)

⇒ (6 - 4a - 7b)2 = 36 + 16a2 - 48a + 49b2 - 14b(6 - 4a)

⇒ (6 - 4a - 7b)2 = 36 + 16a2 + 49b2 - 48a + 56ab - 84b.

Hence, (6 - 4a - 7b)2 = 36 + 16a2 + 49b2 - 48a + 56ab - 84b.

(iii) On solving,

⇒ (7 - 3xy)3 = (7)3 - (3xy)3 - 3(7)(3xy)(7 - 3xy)

⇒ (7 - 3xy)3 = 343 - 27x3y3 - 63xy(7 - 3xy)

⇒ (7 - 3xy)3 = 343 - 27x3y3 - 441xy + 189x2y2.

Hence, (7 - 3xy)3 = 343 - 27x3y3 - 441xy + 189x2y2.

(iv) On solving,

⇒ (x + y + 2)3 = (x)3 + (y + 2)3 + 3(x)(y + 2)(x + y + 2)

⇒ (x + y + 2)3 = x3 + y3 + 23 + 3(y)(2)(y + 2) + (3xy + 6x)(x + y + 2)

⇒ (x + y + 2)3 = x3 + y3 + 8 + 6y(y + 2) + 3xy(x + y + 2) + 6x(x + y + 2)

⇒ (x + y + 2)3 = x3 + y3 + 8 + 6y2 + 12y + 3x2y + 3xy2 + 6xy + 6x2 + 6xy + 12x

⇒ (x + y + 2)3 = x3 + y3 + 3x2y + 3xy2 + 6x2 + 6y2 + 12xy + 12x + 12y + 8.

Hence, (x + y + 2)3 = x3 + y3 + 3x2y + 3xy2 + 6x2 + 6y2 + 12xy + 12x + 12y + 8.

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