Solution 2.1:2b
From Förberedande kurs i matematik 1
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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 | ||
- | <math> | + | {{Displayed math||<math>\begin{align} |
- | + | ||
- | \begin{align} | + | |
(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\\ | ||
- | &=1+15x-5x-75x^2 | + | &=1+15x-5x-75x^2\\ |
- | \end{align} | + | &=1+10x-75x^2\,\textrm{.} |
- | </math> | + | \end{align}</math>}} |
- | 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 | + | 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>, |
- | <math> | + | {{Displayed math||<math>\begin{align} |
- | + | ||
- | \begin{align} | + | |
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)\\ | ||
- | &=12-75x^2 | + | &=12-75x^2\,\textrm{.} |
- | \end{align} | + | \end{align}</math>}} |
- | </math> | + | |
All together, we obtain | All together, we obtain | ||
- | <math> \ | + | {{Displayed math||<math>\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+10x-75x^2-12+75x^2\\ | ||
&= 1-12+10x-75x^2+75x^2\\ | &= 1-12+10x-75x^2+75x^2\\ | ||
- | &=-11+10x | + | &=-11+10x\,\textrm{.} |
- | \end{align} | + | \end{align}</math>}} |
- | </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} |