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Solution 2.1:2b

From Förberedande kurs i matematik 1

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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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\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\\
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&=1+15x-5x-75x^2
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\end{align}
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</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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\qquad
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\begin{align}
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3(2-5x)(2+5x) &= 3\big( 2^2-(5x)^2\big)\\
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&=3(4-25x^2)\\
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&=12-75x^2
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\end{align}
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</math>
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All together, we obtain
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<math> \qquad (1-5x)(1+15x)-3(2-5x)(2+5x) </math>
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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\\
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&= 1-12+10x-75x^2+75x^2\\
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&=-11+10x
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\end{align}
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</math>
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Revision as of 10:58, 13 August 2008

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

(1−5x)(1+15x)=11+115x−5x1−5x15x=1+15x−5x−75x2

As for the second expression, we can use the conjugate rule (a−b)(a+b)=a2−b2 where a=2 and b=5x

3(2−5x)(2+5x)=322−(5x)2=3(4−25x2)=12−75x2

All together, we obtain

(1−5x)(1+15x)−3(2−5x)(2+5x)

=(1+10x−75x2)−(12−75x2)=1+10x−75x2−12+75x2=1−12+10x−75x2+75x2=−11+10x

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