Solution 2.1:2b

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Current revision (08:15, 23 September 2008) (edit) (undo)
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<!--center> [[Image:2_1_2b.gif]] </center-->
 
We expand the first product of bracketed terms by multiplying each term inside the first bracket by each term from the second bracket
We expand the first product of bracketed terms by multiplying each term inside the first bracket by each term from the second bracket
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<math>
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{{Displayed math||<math>\begin{align}
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\qquad
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\begin{align}
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(1-5x)(1+15x) &= 1\cdot 1+1\cdot 15x-5x\cdot 1-5x \cdot 15x\\
(1-5x)(1+15x) &= 1\cdot 1+1\cdot 15x-5x\cdot 1-5x \cdot 15x\\
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&=1+15x-5x-75x^2
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&=1+15x-5x-75x^2\\
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\end{align}
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&=1+10x-75x^2\,\textrm{.}
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</math>
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\end{align}</math>}}
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As for the second expression, we can use the conjugate rule <math>(a-b)(a+b)=a^2-b^2,</math> where <math>a=2</math> and <math> b=5x.</math>
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As for the second expression, we can use the conjugate rule <math>(a-b)(a+b)=a^2-b^2,</math> where <math>a=2</math> and <math> b=5x</math>,
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<math>
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{{Displayed math||<math>\begin{align}
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\qquad
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\begin{align}
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3(2-5x)(2+5x) &= 3\big( 2^2-(5x)^2\big)\\
3(2-5x)(2+5x) &= 3\big( 2^2-(5x)^2\big)\\
&=3(4-25x^2)\\
&=3(4-25x^2)\\
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&=12-75x^2
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&=12-75x^2\,\textrm{.}
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\end{align}
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\end{align}</math>}}
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</math>
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All together, we obtain
All together, we obtain
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<math> \qquad (1-5x)(1+15x)-3(2-5x)(2+5x) </math>
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{{Displayed math||<math>\begin{align}
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(1-5x)(1+15x)-3(2-5x)(2+5x) &= (1+10x-75x^2)-(12-75x^2)\\
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<math>
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\qquad
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\begin{align}
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\phantom{3(2-5x)(2+5x)} &= (1+10x-75x^2)-(12-75x^2)\\
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&= 1+10x-75x^2-12+75x^2\\
&= 1+10x-75x^2-12+75x^2\\
&= 1-12+10x-75x^2+75x^2\\
&= 1-12+10x-75x^2+75x^2\\
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&=-11+10x
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&=-11+10x\,\textrm{.}
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\end{align}
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\end{align}</math>}}
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</math>
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Current revision

We expand the first product of bracketed terms by multiplying each term inside the first bracket by each term from the second bracket

\displaystyle \begin{align}

(1-5x)(1+15x) &= 1\cdot 1+1\cdot 15x-5x\cdot 1-5x \cdot 15x\\ &=1+15x-5x-75x^2\\ &=1+10x-75x^2\,\textrm{.} \end{align}

As for the second expression, we can use the conjugate rule \displaystyle (a-b)(a+b)=a^2-b^2, where \displaystyle a=2 and \displaystyle b=5x,

\displaystyle \begin{align}

3(2-5x)(2+5x) &= 3\big( 2^2-(5x)^2\big)\\ &=3(4-25x^2)\\ &=12-75x^2\,\textrm{.} \end{align}

All together, we obtain

\displaystyle \begin{align}

(1-5x)(1+15x)-3(2-5x)(2+5x) &= (1+10x-75x^2)-(12-75x^2)\\ &= 1+10x-75x^2-12+75x^2\\ &= 1-12+10x-75x^2+75x^2\\ &=-11+10x\,\textrm{.} \end{align}

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