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Mathematics

Using standard formulae, expand each of the following:

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

(ii) (5x+15x)(5x−15x)\Big(5x + \dfrac{1}{5x}\Big)\Big(5x - \dfrac{1}{5x}\Big)

(iii) (2x2+3x2)(2x2−3x2)\Big(2x^2 + \dfrac{3}{x^2}\Big)\Big(2x^2 - \dfrac{3}{x^2}\Big)

Expansions

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Answer

We know that,

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

(i) Given,

⇒ (3a + 2b)(3a - 2b)

⇒ (3a)2 - (2b)2

⇒ 9a2 - 4b2.

Hence, (3a + 2b)(3a - 2b) = 9a2 - 4b2.

(ii) Given,

⇒(5x+15x)(5x−15x)⇒(5x)2−(15x)2⇒25x2−125x2.\Rightarrow \Big(5x + \dfrac{1}{5x}\Big)\Big(5x - \dfrac{1}{5x}\Big)\\[1em] \Rightarrow (5x)^2 - \Big(\dfrac{1}{5x}\Big)^2 \\[1em] \Rightarrow 25x^2 - \dfrac{1}{25x^2}.

Hence, (5x+15x)(5x−15x)=25x2−125x2\Big(5x + \dfrac{1}{5x}\Big)\Big(5x - \dfrac{1}{5x}\Big) = 25x^2 - \dfrac{1}{25x^2}.

(iii) Given,

⇒(2x2+3x2)(2x2−3x2)⇒(2x2)2−(3x2)2⇒4x4−9x4\Rightarrow \Big(2x^2 + \dfrac{3}{x^2}\Big)\Big(2x^2 - \dfrac{3}{x^2}\Big) \\[1em] \Rightarrow (2x^2)^2 - \Big(\dfrac{3}{x^2}\Big)^2 \\[1em] \Rightarrow 4x^4 - \dfrac{9}{x^4} \\[1em]

Hence, (2x2+3x2)(2x2−3x2)=4x4−9x4\Big(2x^2 + \dfrac{3}{x^2}\Big)\Big(2x^2 - \dfrac{3}{x^2}\Big) = 4x^4 - \dfrac{9}{x^4}.

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