Question

Factorisation

$\tt \: 9(2x + 3y) + 12(2x + 3y)(2x - 3y) + 4(2x - 3y)^{2}$
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Answer 1

Correction in the question:

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Factorize 9(2x + 3y)² + 12(2x + 3y)(2x - 3y) + 4(2x - 3y)²

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Solution:

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$\sf \dashrightarrow 9(2x + 3y)^2 + 12(2x + 3y)(2x - 3y) + 4(2x - 3y)^2$

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• We know that (a - b)² = (a - b)(a - b), therefore (2x - 3y)² can be written as (2x - 3y) (2x - 3y)

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$\sf \dashrightarrow 9(2x + 3y)^2 + 12(2x + 3y)(2x - 3y) + \textsf{\textbf{4(2x - 3y)(2x - 3y)}}$

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• 4(2x - 3y) is common in both 12(2x + 3y)(2x - 3y) and 4(2x - 3y)(2x - 3y), So let's take (2x - 3y) out as a common factor and put the two terms inside brackets.

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$\sf \dashrightarrow 9(2x + 3y)^2 + \textsf{\textbf{4(2x - 3y)}}\Big[3(2x + 3y) + (2x - 3y)\Big]$

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• On operating the terms inside the square brackets we get;

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$\sf \dashrightarrow 9(2x + 3y)^2 + 4(2x - 3y)\Big[\textsf{\textbf{6x + 9y + 2x - 3y}}\Big]$

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$\sf \dashrightarrow 9(2x + 3y)^2 + 4(2x - 3y)\Big[\textsf{\textbf{8x + 6y}}\Big]$

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• On operating 9(2x + 3y)² we get;

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$\sf \dashrightarrow 9\Big[\textsf{\textbf{(2x)}}^2 + \textsf{\textbf{(3y)}}^2 + \textsf{\textbf{2(2x)(3y)}} \Big] + 4(2x - 3y)\Big[8x + 6y\Big]$

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$\sf \dashrightarrow 9\Big[\textsf{\textbf{4x}}^2 + \textsf{\textbf{9y}}^2 + \textsf{\textbf{12xy}} \Big] + 4(2x - 3y)\Big[8x + 6y\Big]$

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• On opening the brackets for 9[4x² + 9y² + 12xy] we get;

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$\sf \dashrightarrow 9\Big[\textsf{\textbf{4x}}^2\Big] + 9\Big[\textsf{\textbf{9y}}^2\Big] + 9\Big[\textsf{\textbf{12xy}} \Big] + 4(2x - 3y)\Big[8x + 6y\Big]$

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$\sf \dashrightarrow\textsf{\textbf{36x}}^2 + \textsf{\textbf{81y}}^2 + \textsf{\textbf{108xy}} + 4(2x - 3y)\Big[8x + 6y\Big]$

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• On operating 4(2x - 3y) we get;

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$\sf \dashrightarrow 36x^2 + 81y^2 + 108xy + \Big(\textsf{\textbf{8x - 12y}}\Big)\Big[8x + 6y\Big]$

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• On opening the brackets for (8x - 12y) (8x + 6y) we get;

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$\sf \dashrightarrow 36x^2 + 81y^2 + 108xy + (\textsf{\textbf{8x}})(\textsf{\textbf{8x}}) + (\textsf{\textbf{8x}})(\textsf{\textbf{6y}}) - (\textsf{\textbf{12y}})(\textsf{\textbf{8x}}) - (\textsf{\textbf{12y}})(\textsf{\textbf{6y}})$

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$\sf \dashrightarrow 36x^2 + 81y^2 + 108xy + \textsf{\textbf{64x}}^2 + \textsf{\textbf{48xy}} - \textsf{\textbf{96xy}} - \textsf{\textbf{72y}}^2$

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• On re-arranging based on like terms we get;

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$\sf \dashrightarrow 36x^2 + 64x^2 + 81y^2 - 72y^2 + 108xy + 48xy - 96xy$

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$\sf \dashrightarrow 100x^2 + 9y^2 + 108xy - 48xy$

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$\sf \dashrightarrow 100x^2 + 9y^2 + 60xy$

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$\sf \dashrightarrow 100x^2 + 60xy + 9y^2$

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• 100x² can be written as (10x)²

• 60xy can be written as 2(10x)(3y)

• 9y² can be written as (3y)²

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$\sf \dashrightarrow (10x)^2 + 2(10x)(3y) + (3y)^2$

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• This equation is of the form (a + b)² = a² + b² + 2ab, where a = 10x and b = 3y, therefore it can be written as (10x + 3y)²

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$\sf \dashrightarrow (10x + 3y)^2$

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Hence factored.

Answer 2

Answer:

$answer$

$Factorize 9(2x + 3y)² + 12(2x + 3y)(2x - 3y) + 4(2x - 3y)²</p><p>‎</p><p>Solution:</p><p>‎</p><p>\sf \dashrightarrow 9(2x + 3y)^2 + 12(2x + 3y)(2x - 3y) + 4(2x - 3y)^2⇢9(2x+3y)2+12(2x+3y)(2x−3y)+4(2x−3y)2</p><p>‎</p><p></p><p>We know that (a - b)² = (a - b)(a - b), therefore (2x - 3y)² can be written as (2x - 3y) (2x - 3y)</p><p></p><p>‎</p><p>\sf \dashrightarrow 9(2x + 3y)^2 + 12(2x + 3y)(2x - 3y) + \textsf{\textbf{4(2x - 3y)(2x - 3y)}}⇢9(2x+3y)2+12(2x+3y)(2x−3y)+4(2x - 3y)(2x - 3y)</p><p>‎</p><p></p><p>4(2x - 3y) is common in both 12(2x + 3y)(2x - 3y) and 4(2x - 3y)(2x - 3y), So let's take (2x - 3y) out as a common factor and put the two terms inside brackets.</p><p></p><p>‎</p><p>\sf \dashrightarrow 9(2x + 3y)^2 + \textsf{\textbf{4(2x - 3y)}}\Big[3(2x + 3y) + (2x - 3y)\Big]⇢9(2x+3y)2+4(2x - 3y)[3(2x+3y)+(2x−3y)]</p><p>‎</p><p></p><p>On operating the terms inside the square brackets we get;</p><p></p><p>‎‎</p><p>\sf \dashrightarrow 9(2x + 3y)^2 + 4(2x - 3y)\Big[\textsf{\textbf{6x + 9y + 2x - 3y}}\Big]⇢9(2x+3y)2+4(2x−3y)[6x + 9y + 2x - 3y]</p><p>‎‎</p><p>\sf \dashrightarrow 9(2x + 3y)^2 + 4(2x - 3y)\Big[\textsf{\textbf{8x + 6y}}\Big]⇢9(2x+3y)2+4(2x−3y)[8x + 6y]</p><p>‎‎</p><p></p><p>On operating 9(2x + 3y)² we get;</p><p></p><p>‎‎</p><p>\sf \dashrightarrow 9\Big[\textsf{\textbf{(2x)}}^2 + \textsf{\textbf{(3y)}}^2 + \textsf{\textbf{2(2x)(3y)}} \Big] + 4(2x - 3y)\Big[8x + 6y\Big]⇢9[(2x)2+(3y)2+2(2x)(3y)]+4(2x−3y)[8x+6y]</p><p>‎‎</p><p>\sf \dashrightarrow 9\Big[\textsf{\textbf{4x}}^2 + \textsf{\textbf{9y}}^2 + \textsf{\textbf{12xy}} \Big] + 4(2x - 3y)\Big[8x + 6y\Big]⇢9[4x2+9y2+12xy]+4(2x−3y)[8x+6y]</p><p>‎</p><p></p><p>On opening the brackets for 9[4x² + 9y² + 12xy] we get;</p><p></p><p>‎</p><p>\sf \dashrightarrow 9\Big[\textsf{\textbf{4x}}^2\Big] + 9\Big[\textsf{\textbf{9y}}^2\Big] + 9\Big[\textsf{\textbf{12xy}} \Big] + 4(2x - 3y)\Big[8x + 6y\Big]⇢9[4x2]+9[9y2]+9[12xy]+4(2x−3y)[8x+6y]</p><p>‎</p><p>\sf \dashrightarrow\textsf{\textbf{36x}}^2 + \textsf{\textbf{81y}}^2 + \textsf{\textbf{108xy}} + 4(2x - 3y)\Big[8x + 6y\Big]⇢36x2+81y2+108xy+4(2x−3y)[8x+6y]</p><p>‎</p><p></p><p>On operating 4(2x - 3y) we get;</p><p></p><p>‎‎</p><p>\sf \dashrightarrow 36x^2 + 81y^2 + 108xy + \Big(\textsf{\textbf{8x - 12y}}\Big)\Big[8x + 6y\Big]⇢36x2+81y2+108xy+(8x - 12y)[8x+6y]</p><p>‎‎‎</p><p></p><p>On opening the brackets for (8x - 12y) (8x + 6y) we get;</p><p></p><p>‎‎‎</p><p>\sf \dashrightarrow 36x^2 + 81y^2 + 108xy + (\textsf{\textbf{8x}})(\textsf{\textbf{8x}}) + (\textsf{\textbf{8x}})(\textsf{\textbf{6y}}) - (\textsf{\textbf{12y}})(\textsf{\textbf{8x}}) - (\textsf{\textbf{12y}})(\textsf{\textbf{6y}})⇢36x2+81y2+108xy+(8x)(8x)+(8x)(6y)−(12y)(8x)−(12y)(6y)</p><p>‎‎</p><p>\sf \dashrightarrow 36x^2 + 81y^2 + 108xy + \textsf{\textbf{64x}}^2 + \textsf{\textbf{48xy}} - \textsf{\textbf{96xy}} - \textsf{\textbf{72y}}^2⇢36x2+81y2+108xy+64x2+48xy−96xy−72y2</p><p>‎‎</p><p></p><p>On re-arranging based on like terms we get;</p><p></p><p>‎‎</p><p>\sf \dashrightarrow 36x^2 + 64x^2 + 81y^2 - 72y^2 + 108xy + 48xy - 96xy⇢36x2+64x2+81y2−72y2+108xy+48xy−96xy</p><p>‎‎</p><p>\sf \dashrightarrow 100x^2 + 9y^2 + 108xy - 48xy⇢100x2+9y2+108xy−48xy</p><p>‎</p><p>\sf \dashrightarrow 100x^2 + 9y^2 + 60xy⇢100x2+9y2+60xy</p><p>‎‎‎</p><p>\sf \dashrightarrow 100x^2 + 60xy + 9y^2⇢100x2+60xy+9y2</p><p>‎</p><p></p><p>100x² can be written as (10x)²</p><p></p><p>60xy can be written as 2(10x)(3y)</p><p></p><p>9y² can be written as (3y)²</p><p></p><p>‎</p><p>\sf \dashrightarrow (10x)^2 + 2(10x)(3y) + (3y)^2⇢(10x)2+2(10x)(3y)+(3y)2</p><p>‎</p><p></p><p>This equation is of the form (a + b)² = a² + b² + 2ab, where a = 10x and b = 3y, therefore it can be written as (10x + 3y)²</p><p></p><p>‎</p><p>\sf \dashrightarrow (10x + 3y)^2⇢(10x+3y)2</p><p>‎</p><p>Hence factored.</p><p>$

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