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Factorisation by Regrouping Terms

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Question

Factorize the following expression:
$4 (x y + 1)^{2} - 9 (x - 1)^{2}$

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Solution

Given expression: $4 (x y + 1)^{2} - 9 (x - 1)^{2}$
$= (2)^{2} (x y + 1)^{2} - (3)^{2} (x - 1)^{2}$
$= [2 (x y + 1)]^{2} - [3 (x - 1)]^{2}$
$= (2 x y + 2)^{2} - (3 x - 3)^{2}$
Since, we have the formula
$(a^{2} - b^{2}) = (a + b) (a - b)$
On Substituting $a = (2 x y + 2), b = (3 x - 3)$ in above formula
$\Rightarrow (2 x y + 2)^{2} - (3 x - 3)^{2} = (2 x y + 2 + 3 x - 3) (2 x y + 2 - 3 x + 3)$
$\Rightarrow (2 x y + 2)^{2} - (3 x - 3)^{2} = (2 x y + 3 x - 1) (2 x y - 3 x + 5)$
$∴ 4 (x y + 1)^{2} - 9 (x - 1)^{2} = (2 x y + 3 x - 1) (2 x y - 3 x + 5)$

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Factorisation by Regrouping Terms

Standard VIII Mathematics

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