Question

Factorise the following
9x^2 - 12xy + 4y^2​

Answer 1

$\underline{\underline{\huge{\gray{\tt{\textbf Answer :-}}}}}$

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$\sf\large\bold{\orange{\underline{\blue{ Given :-}}}}$

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• 9x² - 12xy + 4y²

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$\sf\large\bold{\orange{\underline{\blue{ To \: Find :-}}}}$

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• Factorise it

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$\sf\large\bold{\orange{\underline{\blue{ Solution :-}}}}$

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Let us firstly observe an algebraic identity

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➠ (a - b)² = a² + b² - 2ab ⚊⚊⚊⚊ ⓵

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To factorise a expression we firstly need to check weather it could be expressed in form of any algebraic identity

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Going back to the question , 9x² - 12xy + 4y²

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If we look towards the question and ask ,

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• Is 9 the square of any number
• Is 4 the square of any number

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We would get the results as ,

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• 9 the square of 3
• 4 the square of 2

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So , the question can be simplified as ,

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➜ (3x)² - 12xy + (2y)²

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Further the above expressing could be written as ,

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➜ (3x)² - 2(6)xy + (2y)² ⚊⚊⚊⚊ ⓶

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⟮ Comparing ⓵ & ⓶ ⟯

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We get ,

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• a = 3x
• b = 2y
• 2ab = 2(6xy)

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So it can be factorise as ,

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➜ (3x - 2y)²

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Or,

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➨ (3x - 2y)(3x - 2y)

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Additional information

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Some more algebraic identity

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• (a + b)² = a² + 2ab + b²

• (a – b)2 = a² – 2ab + b²

• a² – b² = (a + b)(a – b)

• (x + a)(x + b) = x² + (a + b)x + ab

• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a – b)³ = a³ – b³ – 3ab(a – b)

• a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)

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