Step-by-step explanation:
[tex]1. \: 12 {(a -4 b)}^{2} - 10(a - 4b)[/tex]
[tex] = 2(a - 4b)(6(a - 4b) - 5)[/tex]
[tex] = (2a - 8b)(6a - 24b - 5)[/tex]
[tex]2. \: {x}^{4} - {y}^{4} [/tex]
[tex] = {({x}^{2}) }^{2} - {( {y}^{2}) }^{2} [/tex]
[tex] = ( {x}^{2} + {y}^{2} )( {x}^{2} - {y}^{2} )[/tex]
[tex] = ( {x}^{2} + {y}^{2} )(x + y)(x - y)[/tex]
*******************************
Answer :
[tex]\boxed{ \sf{ \:12 {(a - 4b)}^{2} - 10(a - 4b) = \: 2(a - 4b)(6a - 24b - 5)\: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\sf \: {x}^{4} - {y}^{4} = \: ( {x}^{2} + {y}^{2})( x + y)(x - y) \: }} \\ \\ [/tex]
Step by step calculations :
[tex]\large\underline{\sf{Solution-1}}[/tex]
[tex]\sf \: 12 {(a - 4b)}^{2} - 10(a - 4b) \\ \\ [/tex]
[tex]\sf \: = \: 2.2.3 {(a - 4b)}(a - 4b) - 2.5.(a - 4b) \\ \\ [/tex]
[tex]\sf \: = \: \blue{2}.2.3 \blue{ {(a - 4b)}}(a - 4b) - \blue{ 2}.5. \blue{(a - 4b)} \\ \\ [/tex]
[tex]\sf \: = \: 2(a - 4b)[6(a - 4b) - 5] \\ \\ [/tex]
[tex]\sf \: = \: 2(a - 4b)[6a - 24b - 5] \\ \\ [/tex]
Hence,
[tex]\sf \: 12 {(a - 4b)}^{2} - 10(a - 4b) = \: 2(a - 4b)[6a - 24b - 5] \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
[tex]\sf \: {x}^{4} - {y}^{4} \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: {( {x}^{2} )}^{2} - {( {y}^{2}) }^{2} \\ \\ [/tex]
We know,
[tex]\qquad\boxed{ \sf{ \: {a}^{2} - {b}^{2} = (a + b)(a - b) \: }} \\ \\ [/tex]
So, using this identity, we get
[tex]\sf \: = \: ( {x}^{2} + {y}^{2})( {x}^{2} - {y}^{2}) \\ \\ [/tex]
[tex]\sf \: = \: ( {x}^{2} + {y}^{2})( x + y)(x - y) \\ \\ [/tex]
[tex]\sf \: \bf\implies {x}^{4} - {y}^{4} = \: ( {x}^{2} + {y}^{2})( x + y)(x - y) \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]
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Answers & Comments
Step-by-step explanation:
[tex]1. \: 12 {(a -4 b)}^{2} - 10(a - 4b)[/tex]
[tex] = 2(a - 4b)(6(a - 4b) - 5)[/tex]
[tex] = (2a - 8b)(6a - 24b - 5)[/tex]
[tex]2. \: {x}^{4} - {y}^{4} [/tex]
[tex] = {({x}^{2}) }^{2} - {( {y}^{2}) }^{2} [/tex]
[tex] = ( {x}^{2} + {y}^{2} )( {x}^{2} - {y}^{2} )[/tex]
[tex] = ( {x}^{2} + {y}^{2} )(x + y)(x - y)[/tex]
*******************************
.Hope.It.Helps.
Verified answer
Answer :
[tex]\boxed{ \sf{ \:12 {(a - 4b)}^{2} - 10(a - 4b) = \: 2(a - 4b)(6a - 24b - 5)\: }} \\ \\ [/tex]
[tex]\boxed{ \sf{ \:\sf \: {x}^{4} - {y}^{4} = \: ( {x}^{2} + {y}^{2})( x + y)(x - y) \: }} \\ \\ [/tex]
Step by step calculations :
[tex]\large\underline{\sf{Solution-1}}[/tex]
[tex]\sf \: 12 {(a - 4b)}^{2} - 10(a - 4b) \\ \\ [/tex]
[tex]\sf \: = \: 2.2.3 {(a - 4b)}(a - 4b) - 2.5.(a - 4b) \\ \\ [/tex]
[tex]\sf \: = \: \blue{2}.2.3 \blue{ {(a - 4b)}}(a - 4b) - \blue{ 2}.5. \blue{(a - 4b)} \\ \\ [/tex]
[tex]\sf \: = \: 2(a - 4b)[6(a - 4b) - 5] \\ \\ [/tex]
[tex]\sf \: = \: 2(a - 4b)[6a - 24b - 5] \\ \\ [/tex]
Hence,
[tex]\sf \: 12 {(a - 4b)}^{2} - 10(a - 4b) = \: 2(a - 4b)[6a - 24b - 5] \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
[tex]\sf \: {x}^{4} - {y}^{4} \\ \\ [/tex]
can be rewritten as
[tex]\sf \: = \: {( {x}^{2} )}^{2} - {( {y}^{2}) }^{2} \\ \\ [/tex]
We know,
[tex]\qquad\boxed{ \sf{ \: {a}^{2} - {b}^{2} = (a + b)(a - b) \: }} \\ \\ [/tex]
So, using this identity, we get
[tex]\sf \: = \: ( {x}^{2} + {y}^{2})( {x}^{2} - {y}^{2}) \\ \\ [/tex]
[tex]\sf \: = \: ( {x}^{2} + {y}^{2})( x + y)(x - y) \\ \\ [/tex]
Hence,
[tex]\sf \: \bf\implies {x}^{4} - {y}^{4} = \: ( {x}^{2} + {y}^{2})( x + y)(x - y) \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]