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Factors of Algebraic Expressions and Factorisation

Factorize 9...

Question

Factorize $9 x^{2} - 12 a x - y^{2} - z^{2} - 2 y z + 4 a^{2}$ .

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Solution

We factorize the given expression $9 x^{2} - 12 a x - y^{2} - z^{2} - 2 y z + 4 a^{2}$ as shown below:

$9 x^{2} - 12 a x - y^{2} - z^{2} - 2 y z + 4 a^{2} = (9 x^{2} - 12 a x + 4 a^{2}) - (y^{2} + z^{2} + 2 y z) = [(3 x)^{2} - (2 \times 3 x \times 2 a) + (2 a)^{2}] - (y + z)^{2} (∵ (a + b)^{2} = a^{2} + b^{2} + 2 a b) = (3 x - 2 a)^{2} - (y + z)^{2} (∵ (a - b)^{2} = a^{2} + b^{2} - 2 a b) = (3 x - 2 a + y + z) (3 x - 2 a - y - z) (∵ a^{2} - b^{2} = (a + b) (a - b))$

Hence, $9 x^{2} - 12 a x - y^{2} - z^{2} - 2 y z + 4 a^{2} = (3 x - 2 a + y + z) (3 x - 2 a - y - z)$ .

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