document.write( "Question 318020: Factor the following expression completely. \r
\n" ); document.write( "\n" ); document.write( "x^2 - 8x + 16 - 9y^2 \n" ); document.write( "

Algebra.Com's Answer #227701 by jim_thompson5910(35256) $\"\"$ $\"About$

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First let's focus on $\"x%5E2+-+8x+%2B+16\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-8x%2B16\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-8\"$ , and the last term is $\"16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"16\"$ to get $\"%281%29%2816%29=16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"16\"$ (the previous product) and add to the second coefficient $\"-8\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"16\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"16\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8,-16\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"16\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*16 = 16
\n" ); document.write( "2*8 = 16
\n" ); document.write( "4*4 = 16
\n" ); document.write( "(-1)*(-16) = 16
\n" ); document.write( "(-2)*(-8) = 16
\n" ); document.write( "(-4)*(-4) = 16\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-8\"$ :\r
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First Number	Second Number	Sum
1	16	1+16=17
2	8	2+8=10
4	4	4+4=8
-1	-16	-1+(-16)=-17
-2	-8	-2+(-8)=-10
-4	-4	-4+(-4)=-8

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"-4\"$ add to $\"-8\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"-4\"$ both multiply to $\"16\"$ and add to $\"-8\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-8x\"$ with $\"-4x-4x\"$ . Remember, $\"-4\"$ and $\"-4\"$ add to $\"-8\"$ . So this shows us that $\"-4x-4x=-8x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-4x-4x%29%2B16\"$ Replace the second term $\"-8x\"$ with $\"-4x-4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-4x%29%2B%28-4x%2B16%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-4%29%2B%28-4x%2B16%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-4%29-4%28x-4%29\"$ Factor out $\"4\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-4%29%28x-4%29\"$ Combine like terms. Or factor out the common term $\"x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-4%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-8x%2B16\"$ factors to $\"%28x-4%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2-8x%2B16=%28x-4%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2+-+8x+%2B+16-9y%5E2=%28x-4%29%5E2-9y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"%28x-4%29%5E2-9y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-4%29%5E2-9y%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-4%29%5E2-%283y%29%5E2\"$ Rewrite $\"9y%5E2\"$ as $\"%283y%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Notice how we have a difference of squares $\"A%5E2-B%5E2\"$ where in this case $\"A=x-4\"$ and $\"B=3y\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So let's use the difference of squares formula $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ to factor the expression:\r
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\n" ); document.write( "\n" ); document.write( " $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ Start with the difference of squares formula.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-4%29%5E2-%283y%29%5E2=%28x-4%2B3y%29%28x-4-3y%29\"$ Plug in $\"A=x-4\"$ and $\"B=3y\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So this shows us that $\"%28x-4%29%5E2-9y%5E2\"$ factors to $\"%28x-4%2B3y%29%28x-4-3y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So this then means that $\"x%5E2+-+8x+%2B+16-9y%5E2=%28x-4%2B3y%29%28x-4-3y%29\"$ . \n" ); document.write( "

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